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“ENTROPIC” SOLUTIONS TO A THERMODYNAMICALLY CONSISTENT PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE ELISABETTA ROCCA AND RICCARDA ROSSI

Abstract. In this paper we analyze a PDE system modelling (non-isothermal) phase transitions and damage phenomena in thermoviscoelastic materials. The model is thermodynamically consistent: in particular, no small perturbation assumption is adopted, which results in the presence of quadratic terms on the right-hand side of the temperature equation, only estimated in L1 . The whole system has a highly nonlinear character. We address the existence of a weak notion of solution, referred to as “entropic”, where the temperature equation is formulated with the aid of an entropy inequality, and of a total energy inequality. This solvability concept reflects the basic principles of thermomechanics as well as the thermodynamical consistency of the model. It allows us to obtain global-in-time existence theorems without imposing any restriction on the size of the initial data. We prove our results by passing to the limit in a time discretization scheme, carefully tailored to the nonlinear features of the PDE system (with its “entropic” formulation), and of the a priori estimates performed on it. Our time-discrete analysis could be useful towards the numerical study of this model.

Key words: damage, phase transitions, thermoviscoelasticity, global-in-time weak solutions, time discretization. AMS (MOS) subject classification: 35D30, 74G25, 93C55, 82B26, 74A45. 1. Introduction We consider the following PDE system ϑt + χt ϑ + ρϑ div(ut ) − div(K(ϑ)∇ϑ) = g + a(χ)ε(ut )Vε(ut ) + |χt |2 utt − div(a(χ)Vε(ut ) + b(χ)Eε(u) − ρϑ1) = f in Ω × (0, T ),

in Ω × (0, T ),

χt + µ∂I(−∞,0] (χt ) − div(|∇χ|p−2 ∇χ) + W 0 (χ) 3 −b0 (χ) ε(u)Eε(u) + ϑ in Ω × (0, T ), 2 supplemented with the boundary conditions (here n denotes the outward unit normal to ∂Ω) K(ϑ)∇ϑ · n = h,

∂n χ = 0

u = 0,

on ∂Ω × (0, T ).

(1.1) (1.2) (1.3)

(1.4)

´mond’s modeling approach (see [12, 13]), in [28]. There, Equations (1.1)–(1.3) were derived according to M. Fre it was shown that this PDE system describes (non-isothermal) phase transitions, or (non-isothermal) damage, in a material body occupying a reference domain Ω ⊂ Rd , d ∈ {2, 3}. We refer to [28] for a quite detailed survey on the literature on phase transition and damage problems in thermoviscoelasticity. In (1.1)–(1.3), the symbols ϑ and u respectively denote the absolute temperature of the system and the small displacement vector, while χ is an internal parameter: its meaning depends on the phenomenon described by (1.1)–(1.3), which also determines the choices of the coefficients a and b in the momentum equation (1.2), and of the constant µ ∈ {0, 1} in (1.3). More precisely, Date: 12.03.2014. The work of E.R. was supported by the FP7-IDEAS-ERC-StG Grant #256872 (EntroPhase) and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). R.R. was partially supported by a MIUR-PRIN’10-’11 grant for the project “Calculus of Variations”, and by GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilit` a e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). 1

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ELISABETTA ROCCA AND RICCARDA ROSSI

- the choices a(χ) = 1 − χ and b(χ) = χ correspond to the case of phase transitions in thermoviscoelastic materials: in this case, χ is the order parameter, standing for the local proportion of one of the two phases. We assume that χ takes values between 0 and 1, choosing 0 and 1 as reference values: in the case of phase transitions, χ = 1 stands for the liquid phase while χ = 0 for the solid one and one has 0 < χ < 1 in the so-called mushy regions. Unidirectionality, or irreversibility, of the phase transition process may be encompassed in the model by taking µ = 1 in (1.3), which “activates” the term ∂I(−∞,0] (χt ) (i.e. the subdifferential in the sense of convex analysis of the indicator function I(−∞,0] , evaluated at χt ), yielding the constraint χt ≤ 0 a.e. in Ω × (0, T ). The meaning of a(χ) = 1 − χ and b(χ) = χ in (1.2) is that, in the purely solid phase χ = 0 only the elastic energy, in addition to the thermal expansion energy, contributes to the stress σ = a(χ)Vε(ut ) + b(χ)Eε(u) − ρϑ1 (where E and V are the elasticity and viscosity tensors, respectively). Instead, in the purely liquid, or “viscous”, phase χ = 1 only the viscosity contribution remains, whereas in mushy regions both elastic and viscous effects are present. - The choices a(χ) = b(χ) = χ correspond to damage. In this case, χ is the damage parameter, assessing the soundness of the material microscopically, around a point in the material domain Ω. In fact, we have χ = 0 in the presence of complete damage, while χ takes the value 1 when the material is fully sound, and 0 < χ < 1 describes partial damage. Finally, K in (1.1) is the heat conductivity, W in (1.3) is a mixing energy density, which we assume of the form W = βb + γ b

b → R convex, possibly nonsmooth, and γ with βb : dom(β) b ∈ C2 (R),

while f is a given bulk force, and g and h heat sources. Observe that, in the case when both coefficients a(χ) and b(χ) in the momentum equation degenerate to zero (which happens, for instance, with a(χ) = b(χ) = χ, when complete damage occurs), the equation for u loses its elliptic character. This leads to serious troubles as, for instance, no control of the term b0 (χ) ε(u)Eε(u) 2 on the right-hand side of (1.3) is possible. That is why, in what follows we shall confine our analysis of system (1.1)–(1.3) only to the case case in which the functions a, b ∈ C1 (R) are bounded from below away from 0 (cf. (2.16) in Sec. 2). The reader may refer to our previous contribution [28], where we deal with complete damage and elliptic degeneracy of the momentum equation, in a simplified case. In fact, in [28] we analyzed the following reduced system ϑt + χt ϑ + ρϑ div(ut ) − div(K(ϑ)∇ϑ) = g in Ω × (0, T ), utt − div(a(χ)Vε(ut ) + b(χ)Eε(u) − ρϑ1) = f in Ω × (0, T ),

(1.5)

χt + µ∂I(−∞,0] (χt ) − div(|∇χ|p−2 ∇χ) + W 0 (χ) 3 −b0 (χ) ε(u)Eε(u) + ϑ in Ω × (0, T ), 2 where the quadratic contributions in the velocities on the right-hand side in the internal energy balance (1.1) are neglected by means of the small perturbation assumption (cf. [14]). In this paper, instead, we address the full system (1.1)–(1.3). Let us stress that, since we keep the quadratic terms a(χ)ε(ut )Vε(ut ) and |χt |2 on the right-hand side of (1.1), the model is thermodynamically consistent, as shown in [28]. However, - the highly nonlinear character of the whole system, with the multivalued term ∂I(−∞,0] (χt ) and the possibly nonsmooth contribution βb to the energy W ; - the quadratic terms on the right-hand side of (1.1), which make it difficult to get suitable estimates on (ϑ, u, χ), bring about severe difficulties in the analysis of (1.1)–(1.3). This is the reason why we are going to develop an existence analysis only for a suitable weak solution concept for (1.1)–(1.3), which we illustrate in the following lines. The “entropic” formulation. We resort to a weak solution notion for (1.1)–(1.3) partially mutuated from [9]. There, a thermodynamically consistent model for phase transitions, consisting of the temperature and

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of the phase parameter equations, was analyzed: the temperature equation, featuring quadratic terms on its right-hand side, was weakly formulated in terms of an entropy inequality and of a total energy inequality. In the present framework, the pointwise internal energy balance (1.1) is thus replaced by this entropy inequality Z tZ Z tZ Z tZ K(ϑ)∇ log(ϑ) · ∇ϕ dx dr div(ut )ϕ dx dr − (log(ϑ) + χ)ϕt dx dr + ρ s

s

Ω

ϕ K(ϑ) ∇ log(ϑ) · ∇ϑ dx dr − ϑ Ω

≤− s

Z tZ −

h s

∂Ω

s

Ω

Z tZ

Z tZ s

Ω

g + a(χ)ε(ut )Vε(ut ) + |χt |2

Ω

ϕ dx dr ϑ

(1.6)

ϕ dS dr, ϑ

where ϕ is a sufficiently regular, positive test function (cf. (2.37)), coupled with the following total energy inequality Z tZ Z tZ Z tZ χ χ E (ϑ(t), u(t), ut (t), (t)) ≤ E (ϑ(s), u(s), ut (s), (s))+ g dx dr+ h dS dr+ f ·ut dx dr , (1.7) s

Ω

0

∂Ω

s

Ω

where E (ϑ, u, ut , χ) :=

Z

1 ϑ dx + 2 Ω

Z

1 |ut | dx + 2 Ω 2

Z

1 b(χ(t))ε(u(t))Eε(u(t)) dx + p Ω

Z Ω

|∇χ|p dx +

Z

W (χ) dx . (1.8)

Ω

Both (1.6) and (1.7) are required to hold for almost all t ∈ (0, T ] and almost all s ∈ (0, t), and for s = 0. This formulation of the heat equation has been first developed in [7, 2] in the framework of heat conduction in fluids, ´mond’s approach [12], firstly in [9]. and then applied to a phase transition model, also derived according to Fre Successively, the so-called entropic notion of solution has been used to prove global-in-time existence results in models for special materials like liquid crystals (cf. [8], [10], [11]), and more recently in the analysis of models for the evolution of non-isothermal binary incompressible immiscible fluids (cf. [6]). This notion of solution for the temperature equation corresponds exactly to the physically meaningful requirement that the system should satisfy the second and first principle of Thermodynamics. Indeed, one of the main advantages of this formulation resides in the fact that the thermodynamically consistency of the the model immediately follows from the existence proof. It can be also shown that it is consistent with the standard one, (cf. the discussion in Sec. 2.3, in particular Remark 2.3, and in [9]). From an analytical viewpoint, observe that the entropy inequality (1.6) has the advantage that all the troublesome quadratic terms on the right-hand side of (1.1) feature as multiplied by a negative test function. This, and the fact that (1.6) is an inequality, allows for upper semicontinuity arguments in the limit passage in a suitable approximation of (1.6)–(1.8). In addition to (1.6)–(1.8), the entropic formulation of system (1.1)–(1.3) also consists of the momentum balance (1.2), given pointwise a.e. in Ω×(0, T ), and of the internal variable equation (1.3). The latter is required to hold pointwise almost everywhere in the reversible case µ = 0. In the irreversible case µ = 1, we confine the analysis to the case in which βb is the indicator function I[0,+∞) of [0, +∞), hence W (χ) = I[0,+∞) (χ) + γ b(χ). χ For reasons expounded in Sec. 2.3, we have to weakly formulate (1.3) in terms of the requirement t ≤ 0 a.e. in Ω × (0, T ), of the one-sided variational inequality Z χt − div(|∇χ|p−2 ∇χ) + ξ + γ(χ) + b0 (χ) ε(u)Eε(u) − ϑ ψ dx ≥ 0 for all ψ ∈ W 1,p (Ω) with ψ ≤ 0, 2 Ω (1.9) almost everywhere in (0, T ) (where γ := γ b0 ), and of the energy inequality Z Z tZ 1 χ p 2 χ χ | t | dx dr + |∇ (t)| + W ( (t)) dx p s Ω Ω Z Z tZ 1 χ p 0 χ ε(u)Eε(u) χ χ ≤ |∇ (s)| + W ( (s)) dx + + ϑ dx dr t −b ( ) p 2 s Ω Ω

(1.10)

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ELISABETTA ROCCA AND RICCARDA ROSSI

b χ) = for all t ∈ (0, T ] and almost all s ∈ (0, t), with ξ a selection in the (convex analysis) subdifferential ∂ β( χ ∂I[0,+∞) ( ) of I[0,+∞) . In [28, Prop. 2.14] (see also [16]), we prove that, under additional regularity properties any weak solution in fact fulfills (1.3) pointwise. Let us also mention that other approaches to the weak solvability of coupled PDE systems with an L1 -righthand side are available in the literature: in particular, we refer here to [33] and [30]. In [33], the notion of renormalized solution has been used in order to prove a global-in-time existence result for a nonlinear system in thermoviscoelasticity. In [30] the focus is on rate-independent processes coupled with viscosity and inertia in the displacement equation, and with the temperature equation. There the internal variable equation is not of gradient-flow type as (1.3) but instead features a 1-positively homogeneous dissipation potential. For the resulting PDE system, a weak solution concept partially mutuated from the theory of rate-independent processes by A. Mielke (cf., e.g., [22]) is analyzed. An existence result is proved combining techniques for rate-independent evolution, with Boccardo-Gallou¨et type estimates of the temperature gradient in the heat equation with L1 -right-hand side. Our existence results. The main results of this paper, Theorems 2.5 and 2.8, state the existence of entropic solutions for system (1.1–1.4) in the irreversible (µ = 1) and reversible (µ = 0) cases. More precisely, in the case of unidirectional evolution for χ we can prove the existence of a global-intime entropic solution (i.e. satisfying the entropy (1.6) and the total energy (1.7) inequalities, the (pointwise) momentum balance (1.2), the one-sided (1.9) and the energy (1.10) inequalities for χ). We work under fairly general assumptions on the nonlinear functions in (1.1)–(1.3). More precisely, we require that a and b are sufficiently smooth and bounded from below by a positive constant, b convex, and we standardly assume that W = I[0,+∞) + γ b, with γ b smooth and λ-convex. A crucial role is played by the requirement that the heat conductivity function K = K(ϑ) grows at least like ϑκ with κ > 1, and that the exponent p in the gradient regularization of the equation for χ fulfills p > d. This ensures that χ is estimated in W 1,p (Ω) ⊂ C0 (Ω). Moreover, under some restriction on κ (i.e. κ ∈ (1, 5/3) for space dimension d = 3), we can also obtain an enhanced regularity for ϑ and that conclude that the total energy inequality actually holds as an equality. b In the reversible case (µ = 0), instead, under the same assumptions above described (but with a general β), χ we improve the estimates, hence the regularity, of the internal variable . Therefore, we prove the existence of a weak formulation of (1.1)–(1.3), featuring, in addition to (1.6), (1.7), and (1.2), a pointwise formulation of equation (1.3). Again, in the case of the aforementioned restriction on κ, we enhance the time-regularity of ϑ. What is more, also exploiting the improved formulation of the equation for χ, we are able to conclude existence for a stronger formulation of the heat equation (1.2), of variational type. Instead, a uniqueness result seems to be out of reach, at the moment, not only in the irreversible but also in the reversible cases (cf. Remarks 2.6 and 2.9). Only for the isothermal reversible system a continuous dependence result, yielding uniqueness, can be proved exactly like in [28, Thm.3]. Finally, in the last Section 6 we address the analysis of system (1.1)–(1.3), with µ = 1, in the case the p-Laplacian regularization in (1.3) is replaced by the standard Laplacian operator. We approximate it by adding a p-Laplacian term, modulated by a small parameter δ, on the left-hand side of (1.3), so that Thm. 2.8 guarantees the existence of approximate solutions (ϑδ , uδ , χδ ). Then, we let δ tend to zero. In this context, the enhanced elliptic regularity estimates on the momentum equation exploited in the proof of Thm. 2.5, and which would here yield some suitable compactness for the quadratic term a(χδ )ε(∂t uδ )Vε(∂t uδ ) on the right-hand side of (1.1), are no longer available. In fact, they rely on the requirement p > d. A crucial step for proving the existence of (a slightly weaker notion of) entropic solutions to system (1.1)–(1.3) (cf. Theorem 6.2), then consists in deriving some suitable strong convergence for (∂t uδ )δ with an ad hoc technique, strongly relying on the fact that µ = 1, and on the additional assumption that b is non-decreasing. Our main existence results Thms. 2.5 and 2.8 are proved by passing to the limit in a time-discretization scheme, unique for the reversible and the irreversible cases, carefully tuned to the nonlinear features of the PDE system. In particular, it is devised in such a way as to obtain that the piecewise constant and piecewise linear interpolants of the discrete solutions satisfy the discrete versions of the entropy inequality (1.6), of total energy

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

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inequality (1.8), and of the energy inequality (1.10) in the case µ = 1. Moreover, with delicate calculations we are also able to translate on the time-discrete level a series of a priori estimates on the heat equation, having a nonlinear character. This detailed time-discrete analysis could be interesting in view of further numerical studies of this model. Plan of the paper. In Section 2 we fix some notation, state some preliminaries that will be used in the rest of the paper, list our assumptions on the data as well as our main global-in-time existence results. In Section 3 we perform a series of formal a-priori estimates on our system. We render them rigorously in Section 4, where we set up our time-discrete scheme. Theorems 2.5 and 2.8 are proved by passing to the limit in the approximated entropy and energy inequality, as well as in the discretized versions of (1.2) and (1.3), throughout Sec. 5. Section 6 is then devoted to the proof of Theorem 6.2. 2. Setup and results After fixing some notation and results which shall be used throughout the paper, in Section 2.2 we collect our working assumptions on the nonlinear functions K, a, b, and W in the PDE system (1.1)–(1.3), and on the data. Then, in Secs. 2.3 and 2.4 we discuss the weak formulations of (the initial-boundary value problem for) (1.1)–(1.3) in the irreversible and reversible cases, respectively corresponding to µ = 1 and µ = 0 in (1.3). 2.1. Preliminaries. Notation 2.1. Throughout the paper, given a Banach space X we shall denote by k · kX its norm, and use the symbol h·, ·iX for the duality pairing between X 0 and X. Moreover, we shall denote by BV([0, T ]; X) (by C0weak ([0, T ]; X), respectively), the space of functions from [0, T ] with values in X that are defined at every t ∈ [0, T ] and have bounded variation on [0, T ] (and are weakly continuous on [0, T ], resp.). Let Ω ⊂ Rd be a bounded domain, d ∈ {2, 3}. We set Q := Ω×(0, T ). We identify both L2 (Ω) and L2 (Ω; Rd ) with their dual spaces, and denote by (·, ·) the scalar product in Rd , by (·, ·)L2 (Ω) both the scalar product in L2 (Ω), and in L2 (Ω; Rd ), and by H01 (Ω; Rd ) and H02 (Ω; Rd ) the spaces Z 1 d 1 d 2 H0 (Ω; R ) := {v ∈ H (Ω; R ) : v = 0 on ∂Ω }, endowed with the norm kvkH 1 (Ω;Rd ) := ε(v) : ε(v) dx, 0

H02 (Ω; Rd )

2

Ω

d

:= {v ∈ H (Ω; R ) : v = 0 on ∂Ω }.

Note that k · kH01 (Ω;Rd ) is a norm equivalent to the standard one on H 1 (Ω; Rd ). We will use the symbol D(Q) for the space of the C ∞ -functions with compact support on Q := Ω × (0, T ) and for q > 1 we will adopt the notation W+1,q (Ω) := ζ ∈ W 1,q (Ω) : ζ(x) ≥ 0 for a.a. x ∈ Ω , and analogously for W−1,q (Ω). We denote by Ap the p-Laplacian operator with zero Neumann boundary conditions, viz. Z Ap : W 1,p (Ω) → W 1,p (Ω)0 given by hAp u, viW 1,p (Ω) := |∇u|p−2 ∇u · ∇v dx . Ω

In the weak formulation of the momentum equation (1.2), besides V and E we will also make use of the operator Z 2 −1 d Cρ : L (Ω) → H (Ω; R ) defined by hCρ (θ), viH 1 (Ω;Rd ) := −ρ θ div(v) dx. (2.1) Ω

0

0

Finally, throughout the paper we shall denote by the symbols c, c , C, C various positive constants depending only on known quantities. Furthermore, the symbols Ii , i = 0, 1, ..., will be used as place-holders for several integral terms popping in the various estimates: we warn the reader that we will not be self-consistent with the numbering, so that, for instance, the symbol I1 will occur several times with different meanings. Recaps of mathematical elasticity. The elasticity and viscosity tensors fulfill E = (eijkh ), V = (vijkh ) ∈ C1 (Ω; Rd×d×d×d )

(2.2)

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ELISABETTA ROCCA AND RICCARDA ROSSI

with coefficients satisfying the classical symmetry and ellipticity conditions (with the usual summation convention) eijkh = ejikh = ekhij , vijkh = vjikh = vkhij ∃ α0 > 0 :

eijkh ξij ξkh ≥ α0 ξij ξij

∀ ξij : ξij = ξji

∃ β0 > 0 :

vijkh ξij ξkh ≥ β0 ξij ξij

∀ ξij : ξij = ξji .

(2.3)

Observe that with (2.3) we also encompass in our analysis the case of an anisotropic and inhomogeneous material. In order to give the variational formulation of the momentum equation, we need to introduce the bilinear forms related to the χ-dependent elliptic operators appearing in (1.2). Hence, given a non-negative function η ∈ L∞ (Ω) (later, η = a(χ) or η = b(χ)), let us consider the bilinear symmetric forms e(η·, ·), v(η·, ·) : H01 (Ω; Rd ) × H01 (Ω; Rd ) → R defined for all u, v ∈ H01 (Ω; Rd ) by Z d X e(ηu, v) := h− div(ηEε(u)), viH 1 (Ω;Rd ) = η eijkh εkh (u)εij (v), i,j,k,h=1

v(ηu, v) := h− div(ηVε(u)), viH 1 (Ω;Rd ) =

d X i,j,k,h=1

Ω

(2.4) Z η vijkh εkh (u)εij (v). Ω

Thanks to (2.3) and Korn’s inequality (see eg [4, Thm. 6.3-3]), the forms e(η·, ·) and v(η·, ·) fulfill ( e(ηu, u) ≥ inf x∈Ω (η(x)) C1 kuk2H 1 (Ω) , 1 d ∃ C1 > 0 ∀ u, v ∈ H0 (Ω; R ) : v(ηu, u) ≥ inf x∈Ω (η(x)) C1 kuk2H 1 (Ω) .

(2.5)

It follows from (2.2) that they are also continuous, namely ∃ C2 > 0 ∀ u, v ∈ H01 (Ω; Rd ) :

|e(ηu, v)| + |v(ηu, v)| ≤ C2 kηkL∞ (Ω) kukH 1 (Ω) kvkH 1 (Ω) .

(2.6)

We shall denote by E(η ·) : H01 (Ω; Rd ) → H −1 (Ω; Rd ) and V(η ·) : H01 (Ω; Rd ) → H −1 (Ω; Rd ) the linear operators associated with the forms e(η·, ·) and v(η·, ·), respectively, that is hE (ηv) , wiH 1 (Ω;Rd ) := e(ηv, w),

hV (ηv) , wiH 1 (Ω;Rd ) := v(ηv, w)

for all v, w ∈ H01 (Ω; Rd ).

(2.7)

It can be checked via an approximation argument that the following regularity results hold if η ∈ L∞ (Ω) and u ∈ H01 (Ω; Rd ), then E (ηu) , V (ηu) ∈ H −1 (Ω; Rd ),

(2.8a)

if η ∈ W 1,d (Ω) and u ∈ H02 (Ω; Rd ), then E (ηu) , V (ηu) ∈ L2 (Ω; Rd ).

(2.8b)

Finally, let us recall the following elliptic regularity result (see e.g. [23, Lemma 3.2, p. 260]) ∃ C3 , C 4 > 0

∀ u ∈ H02 (Ω; Rd ) :

C3 kukH 2 (Ω) ≤ k div(ε(u))kL2 (Ω) ≤ C4 kukH 2 (Ω) .

(2.9)

Useful inequalities. In order to make the paper as self-contained as possible, we recall here the GagliardoNirenberg inequality (cf. [24, p. 125]) in a particular case: for all r, q ∈ [1, +∞], and for all v ∈ Lq (Ω) such that ∇v ∈ Lr (Ω), there holds 1 1 1 1 1−θ θ with = θ kvkLs (Ω) ≤ CGN kvkW 1,r (Ω) kvkLq (Ω) − + (1 − θ) , 0 ≤ θ ≤ 1, (2.10) s r d q the positive constant CGN depending only on d, r, q, θ. Combining the compact embedding ( ∞ if d = 2, 2 d 1,d? −η d ? H0 (Ω; R ) b W (Ω; R ), with d = for all η > 0, 6 if d = 3,

(2.11)

(where for d = 2 we mean that H02 (Ω; Rd ) b W 1,q (Ω; Rd ) for all 1 ≤ q < ∞), with [20, Thm. 16.4, p. 102], we have ∀ % > 0 ∃ C% > 0 ∀ u ∈ H02 (Ω; Rd ) : kε(u)kLd? −η (Ω) ≤ %kukH 2 (Ω) + C% kukL2 (Ω) . (2.12)

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

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We will also use the following nonlinear Poincar´e-type inequality (cf. e.g. [15, Lemma 2.2]), with m(w) the mean value of w: ∀q > 0

∃ Cq > 0

∀ w ∈ H 1 (Ω) :

k|w|q wkH 1 (Ω) ≤ Cq (k∇(|w|q w)kL2 (Ω) + |m(w)|q+1 ) .

(2.13)

2.2. Assumptions. In most of this paper, we shall suppose that Ω ⊂ Rd ,

d ∈ {2, 3} is a bounded connected domain, with C2 -boundary ∂Ω.

(2.14)

This will allow us to apply elliptic regularity results and to conclude H 2 (Ω; Rd )-regularity for u. In Section 6 we will see that this regularity requirement can be dropped, at the price of proving the existence of a weaker notion of solution for the irreversible system (cf. Remark 6.3). We list below our basic assumptions on the functions K, a, b, and W in system (1.1)–(1.3). Hypothesis (I). We suppose that the function K : [0, +∞) → (0, +∞) is continuous and κ > 1 ∀ϑ ∈ [0, +∞) : c0 (1 + ϑκ ) ≤ K(ϑ) ≤ c1 (1 + ϑκ ) . Rx b the primitive K(x) b We will denote by K := 0 K(r) dr of K. Hypothesis (II). We require ∃ c0 , c1 > 0

a ∈ C1 (R), b ∈ C2 (R) and ∃ c2 > 0 :

a(x), b(x) ≥ c2 for all x ∈ R

(2.15)

(2.16)

and that the function b is convex. The latter requirement could be weakened to λ-convexity, i.e. that b00 is bounded from below (cf. also (2.19)), see. Remark 4.9 later on. Hypothesis (III). We suppose that the potential W in (1.3) is given by W = βb + γ b, where b → R is proper, l.s.c., convex , βb : dom(β) ∃ cW , c0W > 0 :

W (r) ≥ cW |r| − c0W

γ b ∈ C2 (R),

b . ∀r ∈ dom(β)

(2.17) (2.18)

Moreover, we impose that ∃λ > 0 ∀r ∈ R : γ b00 (r) ≥ −λ.

(2.19)

Hereafter, we shall use the notation b β := ∂ β,

γ := γ b0 .

b ⊂ [0, +∞), which would enforce the (physically feasible) Observe that, we have not required that dom(β) positivity of the phase/damage variable χ. In fact, for the analysis of the irreversible case (i.e. with µ = 1), we will have to confine the discussion to the case βb = I[0,+∞) , cf. Hypothesis (IV) later on. Instead, in the reversible case µ = 0, we will allow for a general βb (complying with Hypothesis (III)). Remark 2.2 (A generalization of the p-Laplacian). In fact, our analysis of system (1.1)–(1.3) extends to the case the p-Laplacian operator −div(|∇χ|p−2 ∇χ), with p > d, is replaced by an elliptic operator B : W 1,p (Ω) → W 1,p (Ω)∗ of the form Z hB(χ), viW 1,p (Ω) := ∇ζ φ(x, ∇χ(x)) · ∇v(x) dx, (2.20) Ω

d

where φ : Ω × R → [0, +∞) is a Carath´eodory integrand such that the map φ(x, ·) : Rd → [0, +∞) is convex, with φ(x, 0) = 0, and in C1 (Rd ) for a.a. x ∈ Ω, φ(x, ζ) ≥ c3 |ζ|p − c4 , ∃ c3 , c4 , c5 > 0 for a.a. x ∈ Ω ∀ ζ ∈ Rd : |∇ζ φ(x, ζ)| ≤ c5 (1 + |ζ|p−1 ) . This more general framework was analyzed in [28], to which we refer the reader for all details.

8

ELISABETTA ROCCA AND RICCARDA ROSSI

Problem and Cauchy data. We suppose that the data f , g, and h fulfill f ∈ L2 (0, T ; L2 (Ω; Rd )), 1

1

1

2

(2.21)

2

1

0

g ∈ L (0, T ; L (Ω)) ∩ L (0, T ; H (Ω) ), h ∈ L (0, T ; L (∂Ω)),

h≥0

g≥0

a.e. in Ω × (0, T ) ,

a.e. in ∂Ω × (0, T ) ,

(2.22) (2.23)

and that the initial data comply with ϑ0 ∈ L1 (Ω),

∃ ϑ∗ > 0 :

u0 ∈ H02 (Ω; Rd ), χ0 ∈ W

1,p

(Ω),

inf ϑ0 ≥ ϑ∗ > 0 , Ω

v0 ∈ L2 (Ω; Rd ) , 1

b χ0 ) ∈ L (Ω). β(

log ϑ0 ∈ L1 (Ω),

(2.24) (2.25) (2.26)

2.3. A global existence result for the irreversible system. Before stating precisely our notion of weak solution to (the initial-boundary value problem for) system (1.1)–(1.3) in the case of unidirectional evolution, let us briefly motivate the weak formulations for the heat balance equation (1.1), and for the phase/damage parameter subdifferential inclusion (1.3) (with µ = 1). They will be coupled with the pointwise (in time and space) formulation of the momentum equation (1.2) (cf. (2.40) later on). For (1.1), we adopt the weak formulation of proposed in [2, 7, 9]. It consists of a so-called “entropy inequality”, and of an “energy conservation” identity. The former is obtained by formally dividing (1.1) by ϑ, and testing it by a smooth test function ϕ. Integrating over space and time leads to Z TZ Z TZ χ K(ϑ)∇ log(ϑ)∇ϕ dx dt (∂t log(ϑ) + t + ρdiv(ut )) ϕ dx dt + 0

0

Ω

Z

Ω T

Z

ϕ K(ϑ) ∇ log(ϑ)∇ϑ dx dt ϑ 0 Ω Z TZ Z TZ ϕ ϕ h dS dt = (g + a(χ)ε(ut )Vε(ut ) + |χt |2 ) dx dt + ϑ 0 ∂Ω ϑ 0 Ω (2.27) for all ϕ ∈ D(Q). Then, the entropy inequality (2.37) below follows. The total energy identity (2.38) associated with system (1.1)–(1.3) is obtained by testing (1.1) by 1, (1.2) by ut , and (1.3) by χt . −

Remark 2.3. Conversely, it can be checked that, when the functions ϑ and χ are sufficiently smooth, inequalities (2.37)–(2.38), combined with (1.2) and (1.3), yield the heat equation (1.1). Indeed, the weak formulation of (1.1) is equivalent, for sufficiently smooth solutions, to the (2.37) with identity sign. Hence, let us suppose, by contradiction, that (2.37) holds with strict inequality sign (hence, (1.1) does not hold). Then, using (1.2) and (1.3), we can conclude that the total energy balance (2.38) is not satisfied. However, at the moment the necessary regularity for ϑ and χ to carry out this argument is out of reach. Let us emphasize that the entropy inequality (2.37) below has the advantage that all the troublesome quadratic quantities on the right-hand side of (1.1) are tested by the negative function −ϕ. This will allow for upper semicontinuity arguments in the limit passage for proving the existence of weak solutions, cf. Sec. 5 later on. Let us also mention in advance that, when dropping the unidirectionality constraint (i.e., in the case µ = 0), under an additional condition (cf. Hypothesis (V)), we will be able to get an existence result for an improved formulation of (1.1), cf. Theorem 2.8 below. A significant difficulty in the analysis of system (1.1)–(1.3) is due to the triply nonlinear character of (1.3), featuring, in addition to the p-Laplacian and to β = ∂ βb which contributes to W 0 , the (maximal monotone) operator ∂I(−∞,0] . Since the latter is unbounded, it is not possible to perform comparison estimates in (1.3) and an estimate for the terms Ap χ and β(χ) (treated as single-valued in the context of this heuristical discussion) could be obtained only by testing (1.3) by ∂t (Ap χ + β(χ)). However, the related calculations, involving an

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

9

integration by parts in time on the right-hand side of (1.3), cannot be carried out in the present case. That is why, we need to resort to a weak formulation of (1.3) which does not feature the term Ap χ + β(χ). We draw it from [16, 17], and as therein we confine the analysis to the particular case in which Hypothesis (IV). βb = I[0,+∞) . (2.28) This still ensures the constraint χ ∈ [0, 1]

a.e. in Ω × (0, T )

(2.29)

provided we start from an initial datum χ0 ≤ 1 a.e. in Ω, we will obtain by irreversibility that χ(t) ≤ χ0 ≤ 1 a.e. in Ω, for almost all t ∈ (0, T ). To motivate the weak formulation of (1.3) from [16, 17], we observe that (1.3) rephrases as χt ≤ 0 in Ω × (0, T ), χt − div(|∇χ|p−2 ∇χ) + ξ + γ(χ) + b0 (χ) ε(u)Eε(u) − ϑ ψ ≥ 0 for all ψ ≤ 0 in Ω × (0, T ), 2 χt − div(|∇χ|p−2 ∇χ) + ξ + γ(χ) + b0 (χ) ε(u)Eε(u) − ϑ χt ≤ 0 in Ω × (0, T ), 2

(2.30a) (2.30b) (2.30c)

with ξ ∈ ∂I[0,+∞) (χ) in Ω × (0, T ). Our weak formulation of (1.3) in fact consists of (2.30a), of the integrated version of (2.30b), with negative test functions from W 1,p (Ω), and of the energy inequality obtained by integrating (2.30c). In [28, Prop. 2.14] (see also [16, Thm. 4.6]), we prove that, under additional regularity properties, any weak solution in the sense of (2.41)–(2.44) in fact fulfills (1.3) pointwise. We are now in the position to specify our weak solution concept, for which we borrow the terminology from [9]. Definition 2.4 (Entropic solutions to the irreversible system). Let µ = 1. Given initial data (ϑ0 , u0 , v0 , χ0 ) fulfilling (2.24)–(2.26), we call a triple (ϑ, u, χ) an entropic solution to the (initial-boundary value problem) for system (1.1)–(1.3), with the boundary conditions (1.4), if ϑ ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (0, T ; L1 (Ω)), 2

(2.31)

1

log(ϑ) ∈ L (0, T ; H (Ω)),

(2.32)

u ∈ H 1 (0, T ; H02 (Ω; Rd )) ∩ W 1,∞ (0, T ; H01 (Ω; Rd )) ∩ H 2 (0, T ; L2 (Ω; Rd )) , ∞

χ ∈ L (0, T ; W

1,p

1

2

(Ω)) ∩ H (0, T ; L (Ω)),

(2.33) (2.34)

(ϑ, u, χ) complies with the initial conditions u(0, x) = u0 (x), ut (0, x) = v0 (x) χ(0, x) = χ0 (x)

for a.a. x ∈ Ω,

(2.35)

for a.a. x ∈ Ω,

(2.36)

(while the initial condition for ϑ is implicitly formulated in (2.38) below), and with the entropic formulation of (1.1)–(1.3), consisting of - the entropy inequality for almost all t ∈ (0, T ] and almost all s ∈ (0, t), and for s = 0: Z tZ Z tZ Z tZ χ (log(ϑ) + )ϕt dx dr − ρ div(ut )ϕ dx dr − K(ϑ)∇ log(ϑ) · ∇ϕ dx dr s

Ω

s

Ω

s

Ω

Z tZ ≤ hlog(ϑ(t)), ϕ(t)iW 1,d+ (Ω) − hlog(ϑ(s)), ϕ(s)iW 1,d+ (Ω) − Z tZ − s

Ω

ϕ g + a(χ)ε(ut )Vε(ut ) + |χt |2 dx dr − ϑ 0

for all ϕ in C ([0, T ]; W

1,d+

Z tZ h s

∂Ω

s

ϕ K(ϑ) ∇ log(ϑ) · ∇ϑ dx dr ϑ Ω

ϕ dS dr ϑ

(Ω)) for some > 0, and ϕ ∈ H 1 (0, T ; L6/5 (Ω)), with ϕ ≥ 0;

(2.37)

10

ELISABETTA ROCCA AND RICCARDA ROSSI

- the total energy inequality for almost all t ∈ (0, T ] and almost all s ∈ (0, t), and for s = 0: Z tZ Z tZ Z tZ f · ut dx dr , h dS dr + g dx dr + E (ϑ(t), u(t), ut (t), χ(t)) ≤ E (ϑ(s), u(s), ut (s), χ(s)) + s

s

Ω

∂Ω

s

Ω

(2.38) where for s = 0 we read ϑ0 , and Z Z Z Z 1 1 1 2 p χ χ χ E (ϑ, u, ut , ) := ϑ dx + |ut | dx + e(b( (t))u(t), u(t)) + |∇ | dx + W (χ) dx ; 2 Ω 2 p Ω Ω Ω

(2.39)

- the momentum equation utt + V (a(χ)ut ) + E (b(χ)u) + Cρ (ϑ) = f

a.e. in Ω × (0, T );

(2.40)

- the weak formulation of (1.3), viz. χt (x, t) ≤ 0 for a.a. (x, t) ∈ Ω × (0, T ), (2.41) Z χt (t)ψ + |∇χ(t)|p−2 ∇χ(t) · ∇ψ + ξ(t)ψ + γ(χ(t))ψ + b0 (χ(t)) ε(u(t))Eε(u(t)) ψ − ϑ(t)ψ dx ≥ 0 2 (2.42) Ω 1,p for all ψ ∈ W− (Ω), for a.a. t ∈ (0, T ), where ξ ∈ ∂I[0,+∞) (χ) in the sense that ξ ∈ L1 (0, T ; L1 (Ω))

and

hξ(t), ψ − χ(t)iW 1,p (Ω) ≤ 0 ∀ ψ ∈ W+1,p (Ω), for a.a. t ∈ (0, T ),

as well as and the energy inequality for all t ∈ (0, T ], for s = 0, and for almost all 0 < s ≤ t Z Z tZ 1 χ p |χt |2 dx dr + |∇ (t)| + W (χ(t)) dx p Ω s Ω Z tZ Z 1 χ χt −b0 (χ) ε(u)Eε(u) + ϑ dx dr. |∇ (s)|p + W (χ(s)) dx + ≤ p 2 s Ω Ω

(2.43)

(2.44)

We now state our existence result for system (1.1)–(1.3) in the case µ = 1. As far as the time-regularity of ϑ goes, observe that we will just prove BV-in-time regularity for log(ϑ) (cf. (2.46) below). Indeed, we will obtain BV-in-time regularity for ϑ, as well, under an additional restriction on the exponent κ in Hypothesis (I) (note that the range of the admissible values below depends on the space dimension), viz. Hypothesis (V). The exponent κ in (2.15) satisfies κ ∈ (1, 5/3)

if d = 3 and κ ∈ (1, 2)

if d = 2 .

(2.45)

Theorem 2.5 (Existence of entropic solutions, µ = 1). Let µ = 1. Assume Hypotheses (I)–(III) and, in addition, (IV) (i.e., βb = I[0,+∞) ), as well as conditions (2.21)–(2.26) on the data f , g, h, ϑ0 , u0 , v0 , χ0 . Then, there exists an entropic solution (in the sense of Definition 2.4) (ϑ, u, χ) to the initial-boundary value problem for system (1.1)–(1.3), such that log(ϑ) ∈ BV([0, T ]; W 1,d+ (Ω)∗ )

for all > 0,

(2.46)

and ξ in (2.43) is given by + ε(u(x, t))E(x)ε(u(x, t)) 0 χ χ ξ(x, t) = −Iχ=0 (x, t) γ( (x, t)) + b ( (x, t)) − ϑ(x, t) , (2.47) 2 for almost all (x, t) ∈ Ω × (0, T ), where Iχ=0 denotes the characteristic function of the set {χ = 0}, and such that ∃ ϑ > 0 such that ϑ(x, t) ≥ ϑ > 0 for a.a. (x, t) ∈ Ω × (0, T ). (2.48) Furthermore, if in addition K satisfies Hypothesis (V), there holds ϑ ∈ BV([0, T ]; W 2,d+ (Ω)∗ )

for every > 0,

and the total energy inequality (2.38) holds for all t ∈ [0, T ], for s = 0, and for almost all s ∈ (0, t).

(2.49)

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11

Observe that (2.49) yields that there exists D ⊂ [0, T ], at most infinitely countable, such that ϑ ∈ C0 ([0, T ] \ D; W 2,d+ (Ω)∗ ). We will develop the proof in Section 5, by passing to the limit in the time-discretization scheme carefully devised in Section 4. Remark 2.6 (Uniqueness and extensions). (1) Uniqueness of solutions for the irreversible system, even in the isothermal case, is still an open problem. This is mainly due to the doubly nonlinear character of (1.3) (cf. also [5] for non-uniqueness examples for a general doubly nonlinear equation). (2) Theorem 2.5 could be easily extended to the case in which the indicator function I(−∞,0] in (1.3) is replaced by α b : R → [0, +∞]

convex, 1-positively homogeneous, with dom(b α) ⊂ (−∞, 0] and 0 ∈ α(0).

(2.50)

2.4. A global existence result for the reversible system. In the case µ = 0, we are able to cope with a weak solvability notion for system (1.1)–(1.3) stronger than the one from Definition 2.4. Indeed, it features a pointwise formulation for the internal parameter equation (1.3), while keeping the entropic formulation for the heat equation (1.1). Under the additional Hypothesis (V), we will also improve the weak formulation of the heat equation (cf. (2.54) below). As a byproduct, we will manage to prove the total energy identity for all t ∈ [0, T ]. Definition 2.7 (Entropic solutions to the reversible system). Let µ = 0. Given initial data (ϑ0 , u0 , v0 , χ0 ) fulfilling (2.24)–(2.26), we call a triple (ϑ, u, χ) an entropic solution to the (initial-boundary value problem) for system (1.1)–(1.3), with the boundary conditions (1.4), if it has the regularity (2.31)–(2.34), it complies with the initial conditions (2.35)–(2.36), and with -

the the the the

entropy inequality (2.37); total energy inequality (2.38) for almost all t ∈ (0, T ], for s = 0, and for almost all s ∈ (0, t); momentum equation (2.40); internal parameter equation χt + Ap χ + ξ + γ(χ) = −b0 (χ) ε(u)Eε(u) + ϑ 2

a.e. in Ω × (0, T ),

(2.51)

with ξ ∈ L2 (0, T ; L2 (Ω)) s.t.

ξ(x, t) ∈ β(χ(x, t)) for a.a. (x, t) ∈ Ω × (0, T ).

(2.52)

Our second main result states the existence of an entropic solution (ϑ, u, χ) (in the sense of Definition 2.7) to the PDE system (1.1)–(1.3). Furthermore, we show that, under the additional Hypothesis (V), the formulation of the heat equation (1.1) improves to a standard variational formulation (cf. (2.54) below), albeit with suitably smooth test functions, and the total energy inequality (2.38) holds as an equality. We shall refer to the solutions thus obtained as weak. Theorem 2.8 (Existence of entropic and weak solutions, µ = 0). Let µ = 0. Assume Hypotheses (I)–(III) and conditions (2.21)–(2.26) on the data f , g, h, ϑ0 , u0 , v0 , χ0 ,. Then, there exists an entropic solution (in the sense of Definition 2.7) (ϑ, u, χ) to the initial-boundary value problem for system (1.1)–(1.3), such that the strict positivity property (2.48) holds for ϑ, and such that χ has the enhanced regularity χ ∈ L2 (0, T ; W 1+σ,p (Ω))

for all 0 < σ
0. (2.54) In this case, the triple (ϑ, u, χ) complies with the total energy equality Z tZ Z tZ χ χ E (ϑ(t), u(t), ut (t), (t)) = E (ϑ(s), u(s), ut (s), (s)) + g dx dr + s

Ω

0

Z tZ f · ut dx dr ,

h dS dr +

∂Ω

s

Ω

(2.55) for all 0 ≤ s ≤ t ≤ T . The proof will be given in Section 5, passing to the limit in the time-discretization scheme set up in Sec. 4. We mention in advance that the argument for (2.54) and for the total energy identity (2.55) for all t ∈ [0, T ] relies on obtaining, for the sequence (uk , χk ) of approximate solutions, the strong convergences uk → u

in H 1 (0, T ; H01 (Ω; Rd )),

χk → χ

in H 1 (0, T ; L2 (Ω)).

(2.56)

This allows us to pass to the limit on the right-hand side of the approximate version of (2.54). In turn, the proof of (2.56) is based on a lim sup-argument, for which it is essential to have preliminarily obtained the pointwise formulation (2.51) of the equation for χ. This is the reason why we have not been able to obtain the improved formulation (2.54) in the irreversible case µ = 1. Remark 2.9 (Uniqueness in the reversible case). As in the irreversible case, a uniqueness result for the full system seems to be out of reach. Instead, for the isothermal case in [28, Thm. 3] we have proved uniqueness and continuous dependence of the solutions on the data. This result has been obtained in the case the pLaplacian operator −div(|∇χ|p−2 ∇χ) is replaced by an elliptic operator of the type described in Remark 2.2, fulfilling an additional non-degeneracy condition, cf. Hypothesis (VII) in [28]: for instance, we may consider −div((1 + |∇χ|2 )(p−2)/2 ). 3. (Formal) A priori estimates In this section, we perform a series of formal estimates on system (1.1)–(1.3). All of these estimates will be rigorously justified on the time-discrete approximation scheme proposed in Section 4. Yet, we believe that, in order to enhance the readability of the paper, it is worthwhile to develop all the significant calculations on the (easier) time-continuous level. This is especially useful for the Second and the Third a priori estimates, which have a non-standard character and are in fact tailored to handle the quadratic terms on the right-hand side of (1.1). More in detail, we start by showing the strict positivity of the temperature ϑ, via a comparison argument in the same lines as the one for proving positivity in [9, Subsection 4.2.1]. All the ensuing estimates rely on this property, starting from the basic energy estimate (i.e. the one corresponding to the total energy inequality (2.38)). After this, we test (1.1) by ϑα−1 , with α ∈ (0, 1). This enables us somehow to confine the troublesome quadratic terms to the left-hand side. Carefully using the Gagliardo-Nirenberg inequality, we infer a bound for ϑα in L2 (0, T ; H 1 (Ω)). Ultimately, exploiting the fact that the heat flux K controls ϑκ (cf. (2.15)) we conclude an estimate for ϑ in L2 (0, T ; H 1 (Ω)). This done, we are in the position to perform all the remaining estimates, i.e. subtracting the temperature equation tested by 1 from the total energy inequality (2.38); performing an elliptic regularity estimate on the momentum equation (1.3), and comparison estimates in (1.1) and (1.3).

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

13

We mention in advance that, with the exception of the last one, all of the ensuing estimates hold both in the reversible (µ = 0), and in the irreversible (µ = 1) cases. Positivity of ϑ [µ ∈ {0, 1}]. Scooping all the quadratic terms in (1.1) to the right-hand side, we obtain ϑt − div(K(ϑ)∇ϑ) = g + a(χ)ε(ut )Vε(ut ) + |χt |2 − χt ϑ − ρϑdiv(ut ) 1 + |χt |2 − Cϑ2 ≥ −Cϑ2 a.e. in Ω × (0, T ), 2 where we have written (1.1) in a formal way, disregarding the (positive) boundary datum h. Indeed, for the first inequality we have used that V is positive definite, that a is strictly positive, and the fact that ≥ g + c|ε(ut )|2 +

| div(ut )| ≤ c(d)|ε(ut )| a.e. in Ω × (0, T )

(3.1)

with c(d) a positive constant only depending on the space dimension d. The second estimate also relies on the fact that g ≥ 0 a.e. in Ω × (0, T ). Therefore we conclude that v solving the Cauchy problem 1 vt = − v 2 , v(0) = ϑ∗ > 0 2 is a subsolution of (1.1). Hence, a comparison argument yields ϑ(·, t) ≥ v(t) > ϑ∗ > 0

for all t ∈ [0, T ] .

(3.2)

First estimate [µ ∈ {0, 1}]. Test (1.1) by 1, (1.2) by ut , (1.3) by χt and integrate over (0, t), t ∈ (0, T ]. Adding the resulting equations and taking into account cancellations, we obtain Z Z Z Z 1 1 1 |ut (t)|2 dx + e(b(χ(t))u(t), u(t)) + |∇χ(t)|p dx + W (χ(t)) dx ϑ(t) dx + 2 Ω 2 p Ω Ω Ω Z Z Z Z 1 1 1 = ϑ0 dx + W (χ0 ) dx |v0 |2 dx + e(b(χ0 )u0 , u0 ) + |∇χ0 |p dx + (3.3) 2 2 p Ω Ω Ω Ω Z tZ Z tZ Z tZ + g dx ds + h dS ds + f · ut dx ds , 0

Ω

0

∂Ω

0

Ω

viz. the total energy equality (2.55). For (3.3), we have also used the integration-by-parts formula Z t Z Z 1 t 1 1 e(b(χ(t))u(s), ut (s)) ds + b0 (χ)χt ε(u)Eε(u) dx ds = e(b(χ(t))u(t), u(t)) − e(b(χ0 )u0 , u0 ) 2 0 Ω 2 2 0 (3.4) RtR RtR as well as the fact that 0 Ω ∂I(−∞,0] (χt )χt dx ds = 0 Ω I(−∞,0] (χt ) dx ds = 0 (where we have formally written ∂I(−∞,0] (χt ) as a single-valued operator). Using (2.21)–(2.26) for the data f , g, h and the initial data (ϑ0 , u0 , χ0 ), the positivity of ϑ (cf. (3.2)), and the coercivity (2.18) of W (cf. Hypothesis (III)), also in view of the Poincar´e inequality we conclude the following estimate kϑkL∞ (0,T ;L1 (Ω)) + kukW 1,∞ (0,T ;L2 (Ω;Rd )) + kb(χ)1/2 ε(u)kL∞ (0,T ;L2 (Ω;Rd×d )) + kχkL∞ (0,T ;W 1,p (Ω)) ≤ C , (3.5) as well as kW (χ)kL∞ (0,T ;L1 (Ω)) ≤ C .

(3.6) 0

α

α−1

Second estimate [µ ∈ {0, 1}]. Let F (ϑ) = ϑ /α, with α ∈ (0, 1). We test (1.1) by F (ϑ) := ϑ integrate on (0, t) with t ∈ (0, T ]. We thus have Z Z tZ Z tZ Z tZ 0 0 F (ϑ0 ) dx + gF (ϑ) dx ds + hF (ϑ) dS ds + a(χ)ε(ut )Vε(ut )F 0 (ϑ) dx ds Ω

0

Z tZ + 0

Ω

Z tZ + 0

Ω

Ω

|χt |2 F 0 (ϑ) dx ds =

0

∂Ω

0

Z tZ

Z F (ϑ(t)) dx + Ω

K(ϑ)∇ϑ∇(F 0 (ϑ)) dx ds

0

Ω

Ω

χt ϑF 0 (ϑ) dx ds + ρ

Z tZ 0

Ω

ϑ div(ut )F 0 (ϑ) dx ds

, and

14

ELISABETTA ROCCA AND RICCARDA ROSSI

whence (cf. (2.5) and the positivity (2.22) and (2.23) of g and h) Z tZ Z Z Z tZ 4(1 − α) t α/2 2 2 0 |χt |2 F 0 (ϑ) dx ds K(ϑ)|∇(ϑ )| dx ds + c |ε(u )| F (ϑ) dx ds + 2 t α2 0 Ω 0 Ω 0 Ω Z ≤ |F (ϑ0 )| dx + I1 + I2 + I3 , Ω

where we have used (2.5) and (2.16). We estimate Z Z Z 1 1 I1 = max{ϑ(t), 1}α dx ≤ max{ϑ(t), 1} dx ≤ C |F (ϑ(t))| dx ≤ α Ω α Ω Ω R since α < 1 and taking into account the previously obtained (3.5). Analogously we can estimate Ω |F (ϑ0 )| dx; moreover, Z Z Z tZ Z tZ 1 t |χt |2 F 0 (ϑ) dx ds + F 0 (ϑ)ϑ2 dx ds. I2 = |χt ϑF 0 (ϑ)| dx ds ≤ 4 0 Ω 0 Ω 0 Ω Using (2.16) and inequality (3.1), we have that Z tZ Z Z Z tZ 1 t 0 2 0 I3 = |ρ| |ϑ div(ut )F (ϑ)| dx ds ≤ |ε(ut )| F (ϑ) dx ds + C F 0 (ϑ)ϑ2 dx ds 4 0 Ω 0 Ω 0 Ω Z Z Z tZ c2 t ≤ |ε(ut )|2 F 0 (ϑ) dx ds + C F 0 (ϑ)ϑ2 dx ds , 4 0 Ω 0 Ω with c2 from (2.16). All in all, we conclude Z Z Z Z Z Z c2 t 1 t 4(1 − α) t α/2 2 2 0 K(ϑ)|∇(ϑ )| dx ds + |ε(ut )| F (ϑ) dx ds + |χt |2 F 0 (ϑ) dx ds α2 4 0 Ω 2 0 Ω 0 Ω Z tZ ≤C +C ϑα+1 dx ds. 0

(3.7)

Ω

Now, we fix q ≥ 4 and introduce the auxiliary quantity η := max{ϑ, 1}. Observe that η is still in H 1 (Ω), and that, for q sufficiently big (see below) we have α α+1 . ≥ whence η (α+1)/q ≤ η α/2 = w. (3.8) 2 q Therefore, taking into account that Z tZ ZZ Z tZ α/2 2 α/2 2 K(ϑ)|∇(ϑ )| dx ds ≥ c1 |∇(ϑ )| dx ds = c1 |∇w|2 dx ds, 0

{ϑ≥1}

Ω

0

Ω

thanks to (2.15), we infer from (3.7) and (3.8) that Z tZ Z t 2 |∇w| dx ds ≤ C + C kwkqLq (Ω) ds. 0

Ω

(3.9)

0

We now apply the Gagliardo-Nirenberg inequality for d = 3 (for d = 2 even better estimates hold true), yielding r kwkLq (Ω) ≤ c1 k∇wkθL2 (Ω;Rd ) kwk1−θ Lr (Ω) + c2 kwkL (Ω)

(3.10)

with 1 ≤ r ≤ q and θ satisfying 1/q = θ/6 + (1 − θ)/r. Hence θ = 6(q − r)/q(6 − r). Observe that θ ∈ (0, 1) if q < 6 and that, by the way, this restriction on q implies that, for (3.8) we need to have α ∈ [1/2, 1). Plugging the Gagliardo-Nirenberg estimate into (3.9) and using Young’s inequality we ultimately conclude Z Z Z t Z t c t 2q(1−θ)/(2−qθ) |∇w|2 dx ds ≤ C + C kwkLr (Ω) ds + C 0 kwkqLr (Ω) ds . (3.11) 2 0 Ω 0 0 Now, choosing r ≤ 2/α, we have that Z 1/r Z 1/r rα/2 kwkLr (Ω) = η dx ≤ η dx ≤ CkϑkL∞ (0,T ;L1 (Ω)) + |Ω| ≤ C , Ω

Ω

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

15

where the latter inequality is due to estimate (3.5). Combining the above estimate with (3.11) we infer a bound for w = η α/2 in L2 (0, T ; H 1 (Ω)) ∩ L∞ (0, T ; Lr (Ω)). Ultimately, also in view of (3.9), we conclude that kϑα/2 kL2 (0,T ;H 1 (Ω))∩L∞ (0,T ;Lr (Ω)) ≤ C.

(3.12)

Third estimate [µ ∈ {0, 1}]. It follows from (3.7) and (2.15) that Z tZ C≥

K(ϑ)|∇(ϑ 0

α/2

Z tZ

2

)| dx ds ≥ c1

Ω

0

Ω

Z tZ = 0

|ϑκ+α−2 ||∇ϑ|2 dx ds

(3.13)

Ω

Z tZ = 0

ϑκ |∇(ϑα/2 )|2 dx ds

|∇(ϑ(κ+α)/2 )|2 dx ds

Ω

with α ∈ [1/2, 1) arbitrary. From (3.13) and the strict positivity of ϑ (3.2) it follows that Z tZ 0

|∇ϑ|2 dx ds ≤ C,

Ω

provided that κ + α − 2 ≥ 0. Observe that, since κ > 1 we can choose α ∈ [1/2, 1) such that this inequality holds. Hence, taking into account estimate (3.5) and applying Poincar´e inequality, we deduce kϑkL2 (0,T ;H 1 (Ω)) ≤ C.

(3.14)

By interpolation (cf. (2.10)), we also get kϑkLh (Ω×(0,T )) ≤ C

with h = 8/3

if d = 3,

h=3

if d = 2 .

(3.15)

For later use, we also point out that estimates (3.13) and (3.14) yield that k∇ϑ(κ−α)/2 kL2 (0,T ;L2 (Ω)) ≤ C. Combining this with estimate (3.5) and using a nonlinear version of the Poincar´e inequality (cf. e.g. (2.13)), we infer kϑ(κ−α)/2 kL2 (0,T ;H 1 (Ω)) , kϑ(κ+α)/2 kL2 (0,T ;H 1 (Ω)) ≤ C.

(3.16)

Fourth estimate [µ ∈ {0, 1}]. We test (1.1) by 1, integrate over (0, t), and subtract the resulting identity from the total energy balance (3.3). We thus obtain Z tZ Z 1 1 χ p v(a(χ)ut , ut ) ds + e(b(χ(t))u(t), u(t)) + |χt |2 dx ds + |∇ (t)| + W (χ(t)) dx 2 Ω 0 0 Ω Ω p Z Z Z Z tZ 1 1 1 χ p = |u0 |2 dx + e(b(χ0 )u0 , u0 ) + |∇ 0 | + W (χ0 ) dx + ϑ (ρ div ut + χt ) dx ds (3.17) 2 Ω 2 Ω p Ω 0 Ω Z tZ + f ut dx ds.

1 2

Z

|ut (t)|2 dx +

0

Z

t

Ω

Using now (2.25)–(2.26) to estimate the initial data (u0 , χ0 ), (2.21) on f , Hyp. (III) (which in particular yields that W is bounded from below), and combining estimate (3.14) on ϑ with (3.1), we obtain kχt kL2 (Ω×(0,T )) + ka(χ)1/2 ε(ut )kL2 (Ω×(0,T );Rd×d ) ≤ C , whence kut kL2 (0,T ;H01 (Ω;Rd )) ≤ C, by (2.16).

(3.18)

16

ELISABETTA ROCCA AND RICCARDA ROSSI

Fifth estimate [µ ∈ {0, 1}]. We use here the crucial assumption that p > d. We test (1.2) by −div(ε(ut )) and integrate on time (cf. also [28, Sec. 3]). Using the assumption p > d, we can fix ζ > 0 such that p ≥ d + ζ, we get Z tZ E (b(χ)u) · div(ε(ut )) dx ds − 0

Ω

Z tZ

∇b(χ)Eε(u)div(ε(ut )) dx ds −

=− 0

Z tZ 0

Ω

b(χ)div(E(ε(u)))div(ε(ut )) dx ds

Ω

t

Z

k∇b(χ)kLd+ζ (Ω;Rd ) kε(u)kLd? −ζ (Ω;Rd×d ) kdiv(ε(ut ))kL2 (Ω;Rd ) ds

≤C 0

Z

t

kukH 2 (Ω;Rd ) kut kH 2 (Ω;Rd ) ds

+C 0

Z ≤σ 0

t

kut k2H 2 (Ω;Rd )

Z t ds + Cσ kχk2W 1,p (Ω) kuk2H 2 (Ω;Rd ) + kuk2H 2 (Ω;Rd ) ds . 0

Here, d? is from (2.11) and we have exploited inequality (2.12) with a constant σ that we will choose later, and some Cσ > 0. Moreover, we have used that kb(χ)kLd+ζ (Ω) ≤ CkχkW 1,p (Ω) . Furthermore, relying on the elliptic regularity result in (2.9) and on (2.16), we obtain Z tZ Z tZ − V (a(χ)ut )) · div(ε(ut )) dx ds = −∇a(χ)Vε(ut )div(ε(ut )) dx ds (3.19) 0

Ω

0

Ω

Z tZ

a(χ)div(V(ε(ut )))div(ε(ut )) dx ds

− 0

Ω

Z tZ ≥C

Z

2

|div(ε(ut ))| dx ds + I1 ≥ c 0

Ω

0

t

kut k2H 2 (Ω;Rd ) ds + I1 ,

where we get Z t Z χ ∇a( )Vε(ut )div(ε(ut )) dx ds |I1 | = 0

Ω

t

Z

k∇a(χ)kLd+ζ (Ω;Rd ) kε(ut )kLd? −ζ (Ω;Rd×d ) kdiv(ε(ut ))kL2 (Ω;Rd ) ds

≤C Z

0 t

kut k2H 2 (Ω;Rd ) ds + Cδ

≤δ 0

Z ≤δ 0

Z 0

t

kut k2H 2 (Ω;Rd ) ds + Cδ %2

t

k∇a(χ)k2Ld+ζ (Ω;Rd ) kε(ut )k2Ld? −ζ (Ω;Rd×d ) ds Z 0

t

Z

kχk2W 1,p (Ω) kut k2H 2 (Ω;Rd ) ds + Cδ C%

0

t

kχk2W 1,p (Ω) kut k2L2 (Ω;Rd ) ds,

again exploiting (2.12), for some positive constants δ and % that we will choose later and for some Cδ , C% > 0. Moreover, we also have that Z tZ Z t Z t 2 ρ ≤η ∇ϑ · div(ε(u )) dx ds ku k ds + C k∇ϑk2L2 (Ω;Rd ) ds (3.20) 2 d t t η H (Ω;R ) 0

Ω

0

0

holds true for some positive constant η to be fixed later and for some Cη > 0. Collecting (3.19)–(3.20), (3.5) R R d and (3.14), and also using that Ω utt div(ε(ut )) dx = 21 dt |ε(ut )|2 dx , we conclude that Ω Z Z t Z Z 1 1 c t |ε(ut (t))|2 dx + c kut k2H 2 (Ω;Rd ) ds ≤ |ε(v0 )|2 dx + Ckf k2L2 (0,T ;L2 (Ω;Rd )) + kut k2H 2 (Ω;Rd ) ds 2 Ω 2 Ω 2 0 0 Z tZ s + C 1 + ku0 k2H 2 (Ω;Rd ) + kut k2H 2 (Ω;Rd ) dr ds , 0

Rt

RtRs

0

where we have used the fact that 0 kuk2H 2 (Ω;Rd ) ds ≤ ku0 k2H 2 (Ω;Rd ) + 0 0 kut k2H 2 (Ω;Rd ) dr ds and chosen σ, δ, % and η sufficiently small. Taking into account condition (2.21) on f , the assumptions on the initial data

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

17

(2.25), and using a standard Gronwall lemma, we conclude kut kL2 (0,t;H02 (Ω;Rd ))∩L∞ (0,t;H01 (Ω;Rd )) ≤ C.

(3.21)

By comparison in (1.2), taking into account the regularity property (2.8b), we also get kutt kL2 (0,t;L2 (Ω;Rd )) ≤ C.

(3.22)

1,d Sixth estimate [µ ∈ {0, 1}]. We multiply (1.1) by w (Ω)∩L∞ (Ω) (in particular, ϑ , with w a test function in W 1,d+ this is true for w ∈ W (Ω) with > 0). We integrate in space, only. We thus obtain (cf. (2.27)) that Z Z Z Z Z Z ∂t log(ϑ)w dx ≤ Hw dx + K(ϑ) ∇ϑ · ∇w dx + K(ϑ) |∇ϑ|2 w dx + Jw dx + hw dS 2 ϑ Ω ∂Ω Ω Ω Ω Ω ϑ . = I1 + I2 + I3 + I4 + I5

where we have used the place-holders H := −χt − ρdiv(ut ) and J := ϑ1 (g + a(χ)ε(ut )Vε(ut ) + |χt |2 ). Estimate (3.18) yields that kHkL2 (0,T ;L2 (Ω)) ≤ C, therefore |I1 | ≤ H(t)kwkL2 (Ω) with H(t) = kH(·, t)kL2 (Ω) ∈ L2 (0, T ). Analogously, also in view of (2.22) and of (3.2) we have that |I4 | ≤

1 J(t)kwkL∞ (Ω) ϑ∗

with J(t) := kJ(·, t)kL1 (Ω) ∈ L1 (0, T ).

(3.23)

Moreover, |I5 | ≤ kh(t)kL2 (∂Ω) kwkL2 (∂Ω) , with kh(t)kL2 (∂Ω) ∈ L1 (0, T ) thanks to (2.23). Using the growth condition (2.15) for K, we estimate Z Z 1 . |I2 | ≤ C ϑκ−1 |∇ϑ||∇w| dx + C |∇ϑ||∇w| dx = I2,1 + I2,2 . (3.24) Ω Ω ϑ Thanks to the previously proved positivity (3.2), we have I2,2 ≤

C O(t)k∇wkL2 (Ω;Rd ) ϑ∗

with O(t) := k∇ϑ(t)kL2 (Ω;Rd ) ∈ L2 (0, T )

by (3.14). We estimate I2,1 via the H¨ older inequality, taking into account (3.13) and (3.16), whence, for d ∈ {2, 3}, . I2,1 ≤ Ckϑ(κ+α−2)/2 ∇ϑkL2 (Ω;Rd ) kϑ(κ−α)/2 kL6 (Ω) k∇wkL3 (Ω;Rd ) = CO∗ (t)k∇wkL3 (Ω;Rd ) with O∗ (t) := kϑ(t)(κ+α−2)/2 ∇ϑ(t)kL2 (Ω;Rd ) kϑ(t)(κ−α)/2 kL6 (Ω) ∈ L1 (0, T ). Finally, we have Z |I3 | ≤ C

ϑ

κ−2

2

Z

|∇ϑ| |w| dx + C

Ω

Ω

1 . |∇ϑ|2 |w| dx = I3,1 + I3,2 . ϑ2

(3.25)

The positivity property (3.2) again guarantees I3,2 ≤

C O(t)2 kwkL∞ (0,T ) ϑ2∗

with O(t)2 ∈ L1 (0, T )

while, using that ϑκ−2 ≤ cϑκ+α−2 + c0 , we infer Z Z . κ+α−2 2 0 2 I3,2 ≤ kwkL∞ (Ω) c ϑ |∇ϑ| dx + c |∇ϑ| dx = kwkL∞ (Ω) O∗ (t) Ω Ω Z Z κ+α−2 2 0 with O∗ (t) = c ϑ(t) |∇ϑ(t)| dx + c |∇ϑ(t)|2 dx ∈ L1 (0, T ), Ω

(3.26)

Ω

thanks to (3.13) and (3.14). Collecting all of the above calculations, we conclude that k∂t log(ϑ)kL1 (0,T ;(W 1,d (Ω)∩L∞ (Ω))∗ ) ≤ C.

(3.27)

18

ELISABETTA ROCCA AND RICCARDA ROSSI

Seventh estimate [µ ∈ {0, 1}], κ ∈ (1, 5/3) if d = 3 and κ ∈ (1, 2) if d = 2. Assume in addition Hypothesis (V). We multiply (1.1) by a test function w ∈ W 1,∞ (Ω) (which e.g. holds if w ∈ W 2,d+ (Ω) for > 0). By comparison we have Z Z Z Z . ϑt w dx ≤ Lw dx + K(ϑ)∇ϑ · ∇w dx + hw dS = I1 + I2 + I3 , Ω

Ω

∂Ω

Ω

where we have set L = −χt ϑ − ρϑdiv(ut ) + g + a(χ)ε(ut )Vε(ut ) + |χt |2 . Therefore, |I1 | ≤ L(t)kwkL∞ (Ω)

with L(t) := kL(t)kL1 (Ω) ∈ L1 (0, T ),

|I3 | ≤ kh(t)kL2 (∂Ω) kwkL2 (∂Ω) with h ∈ L1 (0, T )

thanks to (3.14), (3.18) and (2.23), respectively. As for I2 , in view of (2.15), taking into account (3.13) and using the H¨ older inequality, we obtain |I2 | ≤ Ckϑ(κ−α+2)/2 kL2 (Ω) kϑ(κ+α−2)/2 ∇ϑkL2 (Ω;Rd ) k∇wkL∞ (Ω;Rd ) + Ck∇ϑkL2 (Ω;Rd ) k∇wkL2 (Ω;Rd ) .

(3.28)

Observe that, since α can be chosen arbitrarily close to 1, in view of estimate (3.15) we have that ϑ(κ−α+2)/2 is bounded in L2 (0, T ; L2 (Ω)) if and only if κ < 35 if d = 3, and κ < 2 if d = 2. Under this restriction on κ, we have that |I2 | ≤ CL∗ (t)k∇wkL∞ (Ω) for some L∗ ∈ L1 (0, T ). Ultimately, we conclude that kϑt kL1 (0,T ;W 1,∞ (Ω)∗ ) ≤ C.

(3.29)

Eighth estimate [µ = 0]. In view of the previously obtained estimates (3.5), (3.14), (3.18), and (3.21), a comparison in equation (1.3) yields that (recall that ξ is a selection in β(χ) a.e. in Ω × (0, T )), kAp (χ) + ξkL2 (0,T ;L2 (Ω)) ≤ C whence, by standard elliptic regularity, kAp (χ)kL2 (0,T ;L2 (Ω)) + kξkL2 (0,T ;L2 (Ω)) ≤ C.

(3.30)

In view of the regularity results [31, Thm. 2, Rmk. 2.5], we finally infer the enhanced regularity (2.53) for χ. Remark 3.1 (The p-Laplacian regularization). A close perusal at the above calculations shows that the fact that p > d for the p-Laplacian term in the χ-equation (1.3) has been used only for carrying out the calculations in the Fifth estimate. All the other estimates do not depend on the condition p > d, and would therefore hold if the operator Ap in (1.3) were replaced by the Laplacian. In turn, the Fifth estimate for u will play a crucial role in the limit passage arguments at the basis of the proofs of Theorems 2.5 and 2.8: it will ensure strong compactness in H 1 (0, T ; H02 (Ω; Rd )) (cf. Lemma 5.1) for the sequences of approximate solutions constructed in Sec. 4. Relying on this, we will be able to pass to the limit with the quadratic term |ε(ut )|2 on the right-hand side of (1.1). Nonetheless, in Sec. 6 we will show that, in the case µ = 1 of unidirectional evolution, it is ultimately possible to drop the constraint p > d and in fact we will obtain an existence result for the entropic formulation of system (1.1)–(1.3), in the case (1.3) simply features the Laplacian (i.e. for p = 2). 4. Time discretization In Section 4.1 we set up a single time discretization scheme for both the irreversible (µ = 1) and for the reversible (µ = 0) systems. We then show in Section 4.2 that the piecewise constant and piecewise linear interpolants of the discrete solutions satisfy a total energy inequality, and the approximate versions of the entropy inequality and of equations (1.2)–(1.3). Finally, in Section 4.3 we rigorously prove the a priori estimates from Section 3 in the time-discrete context. Notation 4.1. In what follows, also in view of the extension (2.50) mentioned at the end of Sec. 2.3, we will use α b and α as place-holders for I(−∞,0] and ∂I(−∞,0] .

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

19

4.1. Setup of the time discretization. We consider an equidistant partition of [0, T ], with time-step τ > 0 and nodes tkτ := kτ , k = 0, . . . , Kτ . In this framework, we approximate the data f , g, and h by local means, i.e. setting for all k = 1, . . . , Kτ Z k Z k Z k 1 tτ 1 tτ 1 tτ fτk := f (s) ds , gτk := g(s) ds , hkτ := h(s) ds . (4.1) τ tk−1 τ tk−1 τ tk−1 τ τ τ Consider the following initial data ϑ0τ := ϑ0 ,

u−1 τ := u0 − τ v0 ,

u0τ := u0 ,

χ0τ := χ0 .

(4.2)

We construct discrete solutions to system (1.1)–(1.3) by solving the following elliptic system, featuring the operator Ak : X → H 1 (Ω)∗ , with Z 1 X = {θ ∈ H (Ω) : K(θ)∇θ · ∇v dx is well defined for all v ∈ H 1 (Ω)}, Ak : X → H 1 (Ω)∗ defined by Ω Z Z k hA (θ), viH 1 (Ω) := K(θ)∇θ · ∇v dx − hkτ v dS . Ω

∂Ω

(4.3) χ0 0 Problem 4.2 (Time discretization of the reversible system, µ ∈ {0, 1}). Starting from (u0τ , u−1 τ , τ , ϑτ ) as in 2 d 1,p k χk Kτ k (4.2), find {ϑτ , uτ , τ }k=1 ⊂ X × H0 (Ω; R ) × W (Ω) fulfilling k χk − χk−1 uτ − uk−1 ϑkτ − ϑk−1 τ τ τ + τ ϑkτ + ρ div ϑkτ + Ak (ϑkτ ) = gτk (4.4) τ τ τ 2 2 k k k k−1 k−1 1/2 χk χk−1 χτ − χk−1 − u u − u u − τ τ τ τ τ τ τ τ k−1 + a(χτ )ε Vε + + in H 1 (Ω)∗ , τ τ τ 2 τ ukτ − 2uk−1 + uk−2 ukτ − uτk−1 τ τ χk−1 + V a( ) + E b(χkτ )ukτ + Cρ (ϑkτ ) = fτk a.e. in Ω, (4.5) τ 2 τ τ χkτ − χk−1 √ χk − χτk−1 ε(uk−1 )Eε(uk−1 ) τ τ τ + τ τ + µζτk + Ap (χkτ ) + ξτk + γ(χkτ ) 3 −b0 (χkτ ) + ϑkτ a.e. in Ω , τ τ 2 (4.6) where I ∈ Rd×d×d×d denotes the identity tensor and ξτk ∈ β(χkτ ) k χτ − χτk−1 k ζτ ∈ α τ

a.e. in Ω,

(4.7)

a.e. in Ω.

(4.8)

Remark 4.3 (Features of the time-discretization scheme). A few observations on Problem 4.2 are in order. First of all, let us point out that the scheme is fully implicit and, in particular, (4.6) is coupled to the system (4.4)–(4.5) by the implicit term ϑkτ on the right-hand side, in view of proving the strict positivity (4.10) below for the discrete temperature ϑkτ . As we will see, our argument for (4.10) is the discrete version of the comparison argument developed at the beginning of Section 3 and strongly relies on the structure of the discrete temperature equation (4.4). However, in the case of unidirectional evolution, we could have decoupled the discrete equation for χ from (4.4)–(4.5), replacing (4.6) by χkτ − χk−1 τ τ

+ µζτk + Ap (χkτ ) + ξτk + γ(χkτ ) 3 −b0 (χkτ )

ε(uk−1 )Eε(uk−1 ) τ τ + ϑk−1 τ 2

a.e. in Ω ,

(4.9)

χk −χk−1 χk −χk−1 and, accordingly, replacing the coupling term τ τ τ ϑkτ on the left-hand side of (4.4) by τ τ τ ϑτk−1 . In Remark 4.5 below, we will show how it is still possible to prove the strict positivity of the discrete temperature for this partially decoupled scheme. 2 √ χk −χk−1 1/2 χk −χk−1 Second, observe that τ 2 τ τ τ appears on the right-hand side of (4.4) and, accordingly, τ τ τ τ features on the left-hand side of (4.6). These terms have been added for technical reasons, related to the proof

20

ELISABETTA ROCCA AND RICCARDA ROSSI

of the discrete version of the total energy inequality (2.38), cf. the text above Proposition 4.8. Clearly, they will disappear when passing to the limit with τ ↓ 0. Because of the implicit character of system (4.4)–(4.6), for the existence proof (cf. Lemma 4.4 below) we shall have to resort to a fixed-point type result from the theory for elliptic systems featuring pseudo-monotone operators, drawn from [29, Chap. II]. Indeed, we will not apply it directly to system (4.4)–(4.6), but to an approximation of (4.4)–(4.6), i.e. system (4.15)–(4.17) below, obtained in the following way. We will need to (1) truncate K, along the lines of [15], in such a way as to have a bounded function in the elliptic operator in the temperature equation (4.4). Therefore, the truncated operator KM , with M a positive parameter, shall be defined on H 1 (Ω) (in place of X), with values in H 1 (Ω)∗ (in place of X ∗ ). Accordingly, we shall truncate all occurrences of ϑ in a quadratic term; (2) following [30], add the higher order terms −νdiv(|ε(ukτ )|η−2 Iε(ukτ )) and ν|χkτ |η−2 χkτ , with ν > 0 and η > 4, on the left-hand sides of (4.5) and (4.6), respectively. Their role is to compensate the quadratic terms on the right-hand side of (4.4). As a result, both for d = 2 and for d = 3 the pseudo-monotone operator by means of which we will rephrase system (4.15)–(4.17) will turn out to be coercive, in its ϑ-component, with respect to the H 1 (Ω)-norm; (3) in the case µ = 1, in order to cope with the (possible) unboundedness of the operator α we will have to replace it with its Yosida-regularization αν (cf. [3]), with ν the same parameter as above. Then, in the proof of Lemma 4.4 we will (1) prove the existence of solutions to the approximate discrete system (4.15)–(4.17); (2) pass to the limit in (4.15)–(4.17) first as the truncation parameter M → ∞ and conclude an existence result for an approximation of system (4.4)–(4.6), still depending on the parameter ν > 0; (3) pass to the limit in this approximate system as ν → 0 and conclude the existence of solutions to (4.4)–(4.6). We postpone to Remark 4.6 some comments on the reason why we need to keep the two limit passages as M → ∞ and ν → 0 distinct. Our existence result for Problem 4.2 reads Lemma 4.4 (Existence for the time-discrete Problem 4.2, µ ∈ {0, 1}). Assume Hypotheses (I)–(III), and assumptions (2.21)–(2.26) on the data f , g, h, ϑ0 , u0 , v0 , χ0 . Then, there exists τ¯ > 0 such that for all τ 0 < τ ≤ τ¯ Problem 4.2, admits at least one solution {(ϑkτ , ukτ , χkτ )}K k=1 . k k χk Kτ Furthermore, any solution {(ϑτ , uτ , τ }k=1 of Problem 4.2 fulfills ϑkτ (x) ≥ ϑ > 0

for a.a. x ∈ Ω

(4.10)

for some ϑ = ϑ(T ). Proof. We split the proof in some steps. Step 1: approximation. As already mentioned, we construct our approximation of system (4.4)–(4.6) by truncating K in (4.4) and the quadratic terms in ϑ, replacing α with its Yosida approximation αν , and adding higher order terms to (4.5) and (4.6). Namely, let   K(−M ) if r < −M, KM (r) := (4.11) K(r) if |r| ≤ M,  K(M ) if r > M and accordingly introduce the operator AkM

1

1

: H (Ω) → H (Ω)

∗

defined by

hAkM (θ), viH 1 (Ω)

Z

Z KM (θ)∇θ · ∇v dx −

:= Ω

Observe that, thanks to (2.15) there still holds KM (r) ≥ c0 for all r ∈ R, and therefore Z hAkM (θ), θiH 1 (Ω) ≥ c0 |∇θ|2 dx for all θ ∈ H 1 (Ω). Ω

hkτ v dS.

(4.12)

∂Ω

(4.13)

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

21

We also introduce the truncation operator TM : R → R   −M TM (r) := r  M

if r < −M, if |r| ≤ M, if r > M.

(4.14)

Furthermore, for a given ν > 0 we denote by αν the Yosida approximation of α with parameter ν. Then, we consider following approximation of system (4.4)–(4.6): k χkτ − χk−1 ϑkτ − ϑk−1 uτ − uk−1 τ τ τ k + TM (ϑτ ) + ρ div TM (ϑkτ ) + AkM (ϑkτ ) = gτk (4.15) τ τ τ 2 2 k k k χτ − χk−1 τ 1/2 χkτ − χk−1 uτ − uk−1 uτ − uk−1 τ τ τ τ k−1 χ + in H 1 (Ω)∗ , + a( τ )ε Vε + τ τ τ 2 τ k k−1 ukτ − 2uk−1 + uk−2 τ τ k−1 uτ − uτ χ + V a( τ ) + E b(χkτ )ukτ + Cρ (TM (ϑkτ )) − νdiv(|ε(ukτ )|η−2 Iε(ukτ )) = fτk 2 τ τ in W 1,η (Ω; Rd )∗ , (4.16) χkτ

χk−1 τ

− τ

+

√

τ

χkτ

χk−1 τ

− τ

+ µαν

χkτ

= −b

0

χk−1 τ

− τ

+ Ap (χkτ ) + ξτk + γ(χkτ ) + ν|χkτ |η−2 χkτ

) )Eε(uk−1 ε(uk−1 τ τ (χkτ ) 2

(4.17) +

TM (ϑkτ )

a.e. in Ω ,

with ξτk ∈ β(χkτ ) a.e. in Ω. Step 2: existence of solutions for the approximate system. Observe that system (4.15)–(4.17) can be recast as ϑkτ + χkτ − χk−1 TM (ϑkτ ) + ρ div ukτ − uk−1 TM (ϑkτ ) + τ AkM (ϑkτ ) τ τ k 2 2 k k χτ − χk−1 uτ − uk−1 uτ − uk−1 τ 3/2 χkτ − χk−1 τ τ τ τ k−1 χ (4.18) Vε −τ − − τ a( τ )ε τ τ τ 2 τ in H 1 (Ω)∗ , ukτ + τ V a(χk−1 )(ukτ − uk−1 ) + τ 2 E b(χkτ )ukτ + τ 2 Cρ (TM (ϑkτ )) − ντ 2 div(|ε(ukτ )|η−2 Iε(ukτ )) τ τ = ϑk−1 + τ gτk τ

= 2uk−1 − uk−2 + τ 2 fτk τ τ χkτ +

√

τ χkτ + µτ αν

χkτ − χk−1 τ τ

in W 1,η (Ω; Rd )∗ ,

(4.19)

+ τ Ap (χkτ ) + τ ξτk + τ γ(χkτ ) + ντ |χkτ |η−2 χkτ − τ TM (ϑkτ ) =

χk−1 τ

+

√

τ χk−1 τ

− τb

0

ε(uk−1 )Eε(uk−1 ) τ τ (χkτ ) 2

(4.20) a.e. in Ω .

Denoting by Rk−1 the operator acting on the unknown (ϑkτ , ukτ , χkτ ) and by Hk−1 the vector of the terms on the r.h.s. of the above equations, we can reformulate system (4.18)–(4.20) in the abstract form Rk−1 (ϑkτ , ukτ , χkτ ) = Hk−1 .

(4.21)

It can be checked that Rk−1 is a pseudo-monotone operator (according to [29, Chap. II, Def. 2.1]) on H 1 (Ω) × W01,η (Ω; Rd ) × H 1 (Ω). In order to check that Rk−1 is coercive on that space, it is sufficient to test (4.18) by ϑkτ , (4.19) by ukτ , (4.20) by χkτ and add the resulting equations. To obtain a bound for kϑkτ kH 1 (Ω) we use that AkM is coercive (cf. (4.13)). The additional terms −νdiv(|ε(ukτ )|η−2 Iε(ukτ )) and ν|χkτ |η−2 χkτ in (4.19) and (4.20) enable us to control the quadratic terms on the right-hand side of (4.18). More in detail, the test of

22

ELISABETTA ROCCA AND RICCARDA ROSSI

(4.18) by ϑkτ gives rise, e.g., to the term I1 := |I1 |

R Ω

a(χk−1 )ε(ukτ )Vε(ukτ )ϑkτ dx, which can be estimated as follows τ

≤ Cka(χτk−1 )kL∞ (Ω) kε(ukτ )k2L4 (Ω;Rd×d ) kϑkτ kL2 (Ω) ≤ 41 kϑkτ k2L2 (Ω) + Ckε(ukτ )k4L4 (Ω;Rd×d ) ≤ 14 kϑkτ k2L2 (Ω) +

ντ 2 k η 4 kε(uτ )kLη (Ω;Rd×d )

+ C,

where the first estimate follows from the H¨ older inequality, the second one from the fact that ka(χk−1 )kL∞ (Ω) ≤ τ k−1 1,p 0 χ C since τ ∈ W (Ω) and a ∈ C (R), and the last one relies on η > 4. Therefore, the right-hand side terms can be absorbed by the left-hand side ones, also resulting from the test of (4.19) by ukτ . With analogous R 1/2 calculations we estimate I2 := Ω (|χkτ |2 + τ 2 |χkτ |2 )ϑkτ dx, exploiting the term ντ |χkτ |η−2 χkτ on the left-hand side of (4.20). Therefore, the Leray-Lions type existence result of [29, Chap. II, Thm. 2.6] applies, yielding the existence of a solution (ϑkτ , ukτ , χkτ ) (whose dependence on the parameters M and ν is not highlighted, for simplicity) to (4.15)–(4.17). Step 3: proof of the strict positivity (4.10). Observe first that, for ϑkτ solving (4.15)–(4.17) the strict positivity (4.10) holds for k = 0 with ϑ := ϑ∗ due to (2.24). In order to prove that ϑkτ ≥ ϑ > 0 a.e. in Ω, for every k ≥ 1, we proceed in the same spirit of the proof of the strict positivity of ϑ in Sec. 3 (cf. also [19, Sec. 5.2]). Namely, we start by deducing from (4.4) that Z k Z Z ϑτ − ϑk−1 τ k k w dx + KM (ϑτ )∇ϑτ ∇w dx ≥ −C (ϑkτ )2 w dx for every w ∈ W+1,2 (Ω), (4.22) τ Ω Ω Ω where C is independent of k. We now consider the decreasing sequence {vk } ⊆ R defined recursively as vk − vk−1 = −Cvk2 , v0 = ϑ∗ > 0 , (4.23) τ where C is the same constant of (4.22). We write now (4.23), adding the term − div(KM (ϑkτ )∇vk ) = 0, in the form Z Z Z 1 k (vk − vk−1 )w dx + KM (ϑτ )∇vk · ∇w dx = −C vk2 w dx for every ww ∈ W+1,2 (Ω). τ Ω Ω Ω Subtracting (4.22) from (4.23) and testing the difference by w = Hε (vk − ϑk ), where   if v ≤ 0  0 Hε (v) =

we obtain, since vk < vk−1 that Z

v/ε   1

if v ∈ (0, ε) if v ≥ ε

(vk − vk−1 ) − (ϑkτ − ϑk−1 ) Hε (vk − ϑkτ ) dx ≤ 0 . τ

(4.24)

Ω

Assume now that ϑk−1 ≥ vk−1 a.e. in Ω (which is true for k = 1). Taking ε & 0, (4.24) yields ϑkτ ≥ vk a.e. τ in Ω, and, by induction, ϑkτ ≥ vk > vKτ a.e. in Ω for every k = 1, . . . , Kτ . We now prove that there exists ϑ > 0 such that vKτ ≥ ϑ a.e. in Ω. To this aim, observe that vKτ rewrites as vKτ = G−1 (G(vKτ )), where Rv G(z) := − z 0 s12 ds is monotonlcally increasing on (0, v0 ], G(0+) = −∞, G(v0 ) = 0, hence, by the mean value theorem, for every k = 1, . . . , Kτ there exists sk ∈ [vk , vk−1 ] such that G(vk ) − G(vk−1 ) 1 1 = G0 (sk ) = 2 ≤ 2 , vk − vk−1 sk vk from which we deduce, using (4.23), G(vk ) − G(vk−1 ) 1 ≤ 2 =⇒ G(vKτ ) ≥ −Cτ Kτ . −Cτ vk2 vk Hence, we get ϑkτ > vKτ = G−1 (G(vKτ )) ≥ G−1 (−Cτ Kτ ) = G−1 (−CT ) =: ϑ(T ).

(4.25)

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

23

Thus, we conclude (4.10) with ϑ = G−1 (−CT ). Step 4: passage to the limit as M → ∞. We now pass to the limit in (4.15)–(4.17) as M → ∞, for ν > 0 fixed. In this framework, we will denote by (ϑM , uM , χM ) the solutions of (4.15)–(4.17), with (ϑk−1 , uk−1 , χk−1 ) τ τ τ given and ν > 0 fixed. First of all, we derive a bunch of estimates for (ϑM , uM , χM )M , holding for constants independent of M > 0 (but possibly depending on τ > 0, as well as on norms of (ϑk−1 , uk−1 , χk−1 )). τ τ τ k−1 k−1 χ χ We test (4.15) by 1, (4.16) by uM − uτ , (4.17) by M − τ , and add the resulting relations. Taking into account all cancellations, conditions (2.21)–(2.26), as well as the fact that the Yosida approximation α bν of α b = I(∞,0] is a positive function, we obtain that ∃C > 0 ∀M > 0 :

kϑM kL1 (Ω) + kuM kH 1 (Ω;Rd ) + ν 1/η kε(uM )kLη (Ω;Rd×d ) + kχM kW 1,p (Ω) ≤ C.

(4.26)

We now test (4.15) by TM (ϑM ). Observing that KM (ϑM )∇ϑM ∇(TM (ϑM )) = K(TM (ϑM ))|∇(TM (ϑM )))|2 ϑM TM (ϑM ) ≥ |TM (ϑM )|2

a.e. in Ω,

we get Z Z Z Z 1 |TM (ϑM )|2 dx + K(TM (ϑM ))|∇(TM (ϑM )))|2 dx ≤ |gτk + ϑk−1 ||T (ϑ )| dx + hkτ |TM (ϑM )| dS M M τ τ Ω Ω Ω Z Z ∂Ω k 2 k + |`τ,M ||TM (ϑM )| dx + |jτ,M ||TM (ϑM )| dx Ω

Ω

(4.27) with the place-holders `kτ,M k jτ,M

χM − χk−1 uM − uk−1 τ τ − ρ div , := − τ τ 2 1/2 χM − χk−1 uM − uk−1 uM − uk−1 τ τ τ +τ :=a(χk−1 )ε Vε + τ τ τ τ 2

2 χM − χk−1 τ . τ

Arguing in the same way as in the proof of [28, Thm. 2] (see also [28, Rmk. 2.10] and [15]), combining the growth condition (2.15) on K with the Poincar´e inequality (2.13), and taking into account estimate (4.26), we deduce that Z ∃ c, C > 0 ∀ M > 0 : K(TM (ϑM ))|∇(TM (ϑM )))|2 dx ≥ ck∇(TM (ϑM )))k2L2 (Ω;Rd ) + kTM (ϑM )kκ+2 L3κ+6 (Ω) − C. Ω

On the other hand, Z

|`kτ,M ||TM (ϑM )|2 dx ≤ k`kτ,M kL2 (Ω) kTM (ϑM )kL3 (Ω) kTM (ϑM )kL6 (Ω)

Ω

c k∇(TM (ϑM )))k2L2 (Ω;Rd ) + CkTM (ϑM )k2L3 (Ω) 4 c ≤ k∇(TM (ϑM )))k2L2 (Ω;Rd ) + CkTM (ϑM )k2L1 (Ω) , 2 where we have used that supM k`kτ,M kL2 (Ω) ≤ C thanks to (4.26). The last inequality follows from the fact that H 1 (Ω) b L3 (Ω) ⊂ L1 (Ω), yielding that for all ρ > 0 there exists Cρ > 0 such that kTM (ϑM )kL3 (Ω) ≤ ρkTM (ϑM )kH 1 (Ω) + Cρ kTM (ϑM )kL1 (Ω) . In the same way, estimate (4.26) ensures that Z k |jτ,M ||TM (ϑM )| dx ≤ CkTM (ϑM )kL2 (Ω) . ≤

Ω

All in all, from (4.27), taking into account (4.26) and conditions (2.22) and (2.23) on g and h, we deduce that ∃C > 0 ∀M > 0 :

kTM (ϑM )kH 1 (Ω) + kTM (ϑM )kL3κ+6 (Ω) ≤ C.

We now introduce the notation SM := {x ∈ Ω : ϑM (x) ≤ M },

OM := Ω \ SM .

(4.28)

24

ELISABETTA ROCCA AND RICCARDA ROSSI

In view of estimate (4.28) we have that Z Z M 3κ+6 1 dx ≤ |TM (ϑM )|3κ+6 dx ≤ C OM

OM

whence

|OM | ≤

C M 3κ+6

→ 0 as M → ∞.

(4.29)

Let us finally test (4.15) by ϑM . Relying on the coercivity (4.13) of AkM and again arguing as in the proof of [28, Thm. 2] we find sup kϑM kH 1 (Ω) + kϑM kL3κ+6 (SM ) ≤ C. (4.30) M >0

Here, we have essentially used the same arguments as for treating (4.27) and estimated the terms involving k `kτ,M and jτ,M by means of (4.26). In the end, it remains to estimate the terms αν ((χM − χk−1 )/τ ), Ap (χM ) and ξM in (4.6). First of all, τ k−1 k−1 k−1 χ χ we may suppose that the terms Ap ( τ ), ξτ ∈ β( τ ) from the previous step are bounded in L2 (Ω) by a constant independent of M . Then, we test (4.6) by (Ap (χM ) − Ap (χk−1 ) + (ξM − ξτk−1 )), thus obtaining τ Z λM (Ap (χM ) − Ap (χk−1 ) + ξM − ξτk−1 ) dx + kAp (χM ) + ξM k2L2 (Ω) τ Ω Z Z . k−1 k−1 χ χ = (Ap ( M ) + ξM )(Ap ( τ ) + ξτ ) dx + µM (Ap (χM ) − Ap (χk−1 ) + ξM − ξτk−1 ) dx = I1 + I2 . τ Ω

Ω

√ )/τ + τ (χM − χk−1 Here, we have used the place-holders λM := (χM − χk−1 )/τ ) and )/τ + αν ((χM − χk−1 τ τ τ k−1 k−1 ε(u )Eε(u ) 0 χ η−2 τ τ χ χ µM := ϑM − b ( M ) − γ( M ) − ν( M ) η. With monotonicity arguments, we see that the first 2 integral on the left-hand side is positive. We estimate 1 1 ) + ξτk−1 k2L2 (Ω) . kAp (χM ) + ξM k2L2 (Ω) + kAp (χk−1 τ 2 2 , and from (4.26) for χM and from (4.30) for ϑM that kµM kL2 (Ω) ≤ C It follows from the estimates on uk−1 , χk−1 τ τ for a constant independent of M > 0. Therefore we have I1 ≤

1 1 kAp (χM ) + ξM k2L2 (Ω) + kAp (χk−1 ) + ξτk−1 k2L2 (Ω) + C . τ 4 4 With this, we conclude that kAp (χM ) + ξM kL2 (Ω) ≤ C for a constant independent of M . By the monotonicity of the operator β (cf., e.g., [1, Lemma 3.3]), we find kAp (χM )kL2 (Ω) ≤ C and kξM kL2 (Ω) ≤ C. Then, a comparison argument in (4.6) yields

χM − χk−1

τ

(4.31) µ αν + kAp (χM )kL2 (Ω) + kξM kL2 (Ω) ≤ C.

2 τ L (Ω) I2 ≤

Standard compactness arguments together with (4.30) imply that there exists ϑ ∈ H 1 (Ω) such that, up to a (not relabeled) subsequence, ( ∞ if d = 2, 1 q ϑM * ϑ in H (Ω), ϑM → ϑ in L (Ω) for all q < (4.32) 6 if d = 3. In particular, ϑM → ϑ in measure. Combining this with (4.29) we infer that TM (ϑM ) → ϑ in measure. Therefore, in view of estimate (4.28) and of the Egorov theorem we ultimately have that ϑ ∈ L3κ+6 (Ω),

TM (ϑM ) * ϑ in H 1 (Ω) ∩ L3κ+6 (Ω),

TM (ϑM ) → ϑ in Lq (Ω) for all 1 ≤ q < 3κ + 6. (4.33)

Therefore, taking into account the growth condition (2.15) for K, we have KM (ϑM ) = K(TM (ϑM )) → K(ϑ) in Lq (Ω) for all 1 ≤ q < 3 +

6 . κ

Combining this with the fact that ∇ϑM * ∇ϑ in L2 (Ω; Rd ), we infer on the one hand that AkM (ϑM ) weakly R ek (ϑ) defined by hA ek (ϑ), vi 1,s converges in the space W 1,s (Ω)∗ to the operator A W (Ω) := Ω K(ϑ)∇ϑ∇v dx − R hk v dx for all v ∈ W 1,s (Ω), for some sufficiently big s > 0. On the other hand, a comparison in (4.15) ∂Ω τ

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

25

ek (ϑ) shows that (AkM (ϑM ))M is bounded in H 1 (Ω)∗ . Therefore, it is not difficult to infer that the operator A extends to H 1 (Ω) and coincides with the operator Ak from (4.3), and that AkM (ϑM ) * Ak (ϑ)

in H 1 (Ω)∗

as M → ∞.

(4.34)

From estimates (4.26) and (4.31) we also deduce that there exist u, χ, ξ and, if µ = 1, ζ such that, up to a subsequence, uM * u in W01,η (Ω; Rd ), χM → χ in W 1,p (Ω) (this follows from the fact that (χM )M is bounded in W 1+σ,p (Ω) for all 0 < σ < p1 by [31, Thm. 2, Rmk. 2.5]), ξM * ξ in L2 (Ω), and, if µ = 1, αν ((χM − χk−1 )/τ ) * ζ in L2 (Ω). By the strong-weak closedness in the sense of graphs of αν (viewed as a τ maximal monotone graph in L2 (Ω) × L2 (Ω)), we infer, in the case µ = 1, that ζ = αν ((χkτ − χk−1 )/τ ) a.e. in τ χ Ω. Analogously, the strong-weak closedness property of β yields that ξ ∈ β( ). Combining this convergences with (4.33)–(4.34) we conclude that the functions ϑ, u, χ, ξ, ζ fulfill a.e. in Ω χ − χk−1 τ τ

+

√ χ − χk−1 ε(uk−1 )Eε(uk−1 ) τ τ τ τ + µαν ((χ − χk−1 )/τ ) + Ap (χ) + ξ + γ(χ) + ν|χ|η−2 χ = −b0 (χ) +ϑ τ τ 2

as well as χ − χk−1 u − uk−1 ϑ − ϑk−1 τ τ τ + ϑ + ρ div ϑ + Ak (ϑ) (4.35) τ τ τ 2 2 χ − χk−1 τ 1/2 χ − χk−1 τ τ k−1 k χ = gτ + a( τ )Λk + + in H 1 (Ω)∗ , (4.36) τ 2 τ u − 2uk−1 + uk−2 u − uk−1 τ τ τ χk−1 + V a( ) + E (b(χ)u) + Cρ (ϑ) − νdiv(Γk ) = fτk in W 1,η (Ω; Rd )∗ , (4.37) τ τ2 τ u −uk−1 u −uk−1 where Λk denotes the weak limit of ε M τ τ Vε M τ τ in L2 (Ω), and Γk stands for the weak limit of |ε(uM )|η−2 Iε(uM ) in Lη/(η−1) (Ω; Rd ). In order to identify them, it is sufficient to test (4.16) by uM and show that Z lim sup h−div(|ε(uM )|η−2 Iε(uM )), uM iW 1,η (Ω;Rd ) = lim sup |ε(uM )|η dx ≤ h−div(Γk ), uiW 1,η (Ω;Rd ) , M →∞

M →∞

Ω

which we can do, exploiting that u solves (4.37). This enables us to conclude that Γk = −div(|ε(u)|η−2 Iε(u)) 1,η d and uM → that u strongly in W (Ω; R ). The latter convergence clearly allows us to conclude that Λk = k−1 k−1 u−uτ u−uτ ε Vε . All in all, (ϑ, u, χ) solves the system τ

τ

χ − χk−1 ϑ − ϑk−1 u − uk−1 τ τ τ + ϑ + ρ div ϑ + Ak (ϑ) = gτk (4.38) τ τ τ 2 u − uk−1 u − uk−1 τ 1/2 χ − χk−1 τ τ τ k−1 χ + a( τ )ε Vε + 1+ in H 1 (Ω)∗ , τ τ 2 τ k−1 u − 2uk−1 + uk−2 τ τ k−1 u − uτ χ + E (b(χ)u) + Cρ (ϑ) − νdiv(|ε(u)|η−2 Iε(u)) = fτk + V a( τ ) τ2 τ (4.39) in W 1,η (Ω; Rd )∗ , (1 +

√

τ)

χ − χk−1 τ τ

+ µαν

χ − χk−1 τ τ

k−1

ε(uτ + Ap (χ) + ξτk + γ(χ) + ν|χ|η−2 χ 3 −b0 (χ)

)Eε(uk−1 ) τ +ϑ 2

a.e. in Ω , (4.40) with ξτk ∈ β(χkτ ) a.e. in Ω. It follows from Step 3 and convergences (4.32) that ϑ also fulfills the strict positivity property (4.10).

26

ELISABETTA ROCCA AND RICCARDA ROSSI

Step 5: passage to the limit as ν → 0. We now pass to the limit in (4.38)–(4.40) as ν → 0. We denote by by (ϑν , uν , χν ) the solutions of (4.38)–(4.40) and, as before, obtain a series of estimates independent of the parameter ν. First, we test (4.38) by 1, (4.39) by uν − uk−1 , (4.40) by χν − χk−1 , and add the resulting relations. We τ τ thus conclude that ∃C > 0 ∀ν > 0 :

kϑν kL1 (Ω) + kuν kH 1 (Ω;Rd ) + ν 1/η kε(uν )kLη (Ω;Rd ) + kχν kW 1,p (Ω) ≤ C.

(4.41)

Second, we test (4.38) by ϑα−1 , with α ∈ (0, 1). With the very same calculations as for the Second a priori ν estimates, cf. also the proof of Prop. 4.10 ahead, we conclude that (cf. (3.7)) that 2 Z Z Z χ Z uν − uτk−1 2 α−1 α−1 ν − χk−1 τ 2 c K(ϑν )|∇ϑα/2 | dx + c ε ϑ ϑ dx + c dx ≤ C + C ϑα+1 dx ν ν ν ν τ τ Ω Ω Ω Ω R (κ+α)/2 2 whence, with the same arguments as throughout (3.8)–(3.13), we arrive at Ω |∇ϑν | dx ≤ C for a constant independent of ν. Then, choosing α ∈ [1/2, 1) such that κ + α ≥ 2, we conclude that kϑν kH 1 (Ω) ≤ C

(4.42)

and, again arguing via the nonlinear Poincar´e inequality, we also have that kϑν(κ+α)/2 kH 1 (Ω) ≤ C . ) Ap (χk−1 τ

(4.43)

and, arguing in the very same way as in Step 4, + ξν − We then test (4.40) by (Ap (χν ) − conclude that

χν − χk−1

τ

+ kAp (χν )kL2 (Ω) + kξν kL2 (Ω) ≤ C. (4.44) µ αν

2 τ L (Ω) ξτk−1 )

We can now pass to the limit in system (4.38) –(4.40) as ν ↓ 0. It follows from the previously proved a priori estimates that, along a (not relabeled) subsequence, uν * u in H01 (Ω; Rd ), χν → χ in W 1,p (Ω), and ϑν * ϑ in H 1 (Ω). Using these convergences, it is not difficult to pass to the limit in (4.39) and conclude that u fulfills (4.5). With the same argument as in Step 4, testing (4.39) by uν we conclude that Z Z lim sup ε(uν )Eε(uν ) dx ≤ ε(u)Eε(u) dx, ν→0

Ω

Ω

yielding that uν → u strongly in H 1 (Ω; Rd ). Therefore, uν − uk−1 uν − uk−1 u − uk−1 u − uk−1 τ τ τ τ k−1 χ a(χk−1 )ε Vε → a( )ε Vε τ τ τ τ τ τ

in L1 (Ω).

(4.45) We use this information to pass to the limit in (4.38). Moreover, estimate (4.43) allows us to conclude that, up (κ+α)/2 (κ+α)/2 → ϑ(κ+α)/2 in L6− (Ω) for all > 0, whence, to a subsequence, ϑν * ϑ(κ+α)/2 in H 1 (Ω), hence ϑν taking into account the growth condition on K, that K(ϑν ) → K(ϑ)

in L3+α/κ− (Ω)

for all > 0.

This allows us to pass to the limit in the term K(ϑν )∇ϑν , tested against v ∈ W 1,s (Ω) for some sufficiently big s > 0. All in all, we infer that (ϑ, u, χ) satisfies (4.4) in some dual space W 1,s (Ω)∗ , such that, also, W 1,s (Ω) ⊂ L∞ in accord with the L1 -convergence (4.45). Finally, we pass to the limit in (4.40). Due to estimate (4.44), we have that there exist ξ ∈ L2 (Ω) and, if µ = 1, ζ ∈ L2 (Ω) such that χν − χk−1 τ αν * ζ, ξν * ξ in L2 (Ω). τ The strong-weak closedness of β yields that ξ ∈ β(χ) a.e. in Ω. In order to show that, in the case µ = 1, ζ ∈ α((χ − χk−1 )/τ ) a.e. in Ω, we show that τ Z χ χk−1 Z χν − χk−1 χν − χk−1 − τ τ τ αν dx ≤ ζ dx lim sup τ τ τ ν↓0 Ω Ω

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

27

and invoke well-knows results from the theory of maximal monotone operators. All in all, we conclude that (ϑ, u, χ) solves system (4.4)–(4.6), where (4.4) is to be understood in W 1,s (Ω)∗ . Step 6: H 2 (Ω; Rd )-regularity for ukτ and conclusion. A comparison argument in (4.5) yields that uk − uk−1 τ V a(χkτ ) τ and E b(χkτ )ukτ are in L2 (Ω; Rd ). τ . From the latter information we now deduce that ukτ ∈ H 2 (Ω; Rd ). Indeed, multiplying E b(χkτ )ukτ = hkτ ∈ L2 (Ω; Rd ) by − div(ε(ukτ ), we get Z

b(χkτ )E| div(ε(ukτ )|2

Ω

Z Z k k k k k dx ≤ ∇b(χτ )Eε(uτ ) div(ε(uτ ) dx + hτ div(ε(uτ ) dx Ω Ω Z 2 div(ε(ukτ ) dx ≤ Cδ + δ Ω

where in the latter estimate δ > 0 is sufficiently small, and we have relied on the fact that kukτ kH 1 (Ω;Rd ) + kχkτ kW 1,p (Ω) ≤ C, combined with assumption (2.16) on b. Also using (2.16) and choosing 0 < δ < C1 c2 (cf. (2.5) and (2.16)), we then infer that Z | div(ε(ukτ )|2 dx ≤ C. Ω

Then, a standard regularity result for elliptic systems with constant coefficients (cf. (2.9)), yields that ukτ ∈ H 2 (Ω; Rd ). In the end, exploiting that that ukτ ∈ H 2 (Ω; Rd ), a comparison argument in the heat equation allows us R to conclude that Ω K(ϑ)∇ϑ · ∇v dx is well defined for all test functions v ∈ H 1 (Ω), hence (4.4) is solved in H 1 (Ω)∗ . Remark 4.5. In the case µ = 1, as mentioned in Remark 4.3, the discrete χ-equation could be decoupled χk −χk−1 from the discrete equations for ϑ and u, cf. (4.9). This would lead to having the term τ τ τ ϑk−1 . The τ argument for the strict positivity of ϑkτ in Step 3 in this case would not go through. Nonetheless, it would be possible to prove that ϑkτ ≥ 0 a.e. in Ω, by testing the discrete heat equation by −(ϑkτ )− , and using that R χkτ −χk−1 τ a.e. in Ω. (−(ϑkτ )− ) dx ≥ 0 since χkτ ≤ χk−1 ϑk−1 τ τ Ω

τ

Remark 4.6. We briefly comment on the reason why we need to perform two distinct passages to the limit in the proof of Lemma 4.4. As the above proof shows, in the passage to limit as ν → 0 we lose the information that the right-hand side of the equation for ϑ is estimated in L2 (Ω). Hence, we need to carry out refined estimates on the ϑ-equation (i.e., testing it by ϑα−1 ), where we fully exploit the growth of K to carry out the related calculations. Clearly, to do so we first have to pass to the limit with the truncation parameter. 4.2. Approximate entropy and total energy inequalities. Preliminarily, we establish the Notation 4.7 (Interpolants and discrete integration-by-parts formula). Hereafter, for a given Banach space τ X and a Kτ -tuple (hkτ )K k=1 ⊂ X, we shall use the short-hand notation Dτ,k (h) :=

hkτ − hk−1 τ , τ

D2τ,k (h) := Dτ,k (Dτ,k (h)) =

hkτ − 2hk−1 + hk−2 τ τ . 2 τ

We recall the well-known discrete by-part integration formula Kτ X k=1

τ Kτ τ Dτ,k (h)vτk = hK − h0τ vτ1 − τ vτ

Kτ X k=2

τ hk−1 Dτ,k (v) τ

k Kτ τ for all {hkτ }K k=1 , {vτ }k=1 ⊂ X.

(4.46)

28

ELISABETTA ROCCA AND RICCARDA ROSSI

We introduce the left-continuous and right-continuous piecewise constant, and the piecewise linear interτ polants of the values {hkτ }K k=1 by   hτ : (0, T ) → X defined by hτ (t) := hkτ ,  hτ : (0, T ) → X defined by hτ (t) := hk−1 , for t ∈ (tk−1 , tkτ ]. τ τ  t−tk−1 tk k τ τ −t k−1  hτ : (0, T ) → X defined by hτ (t) := τ hτ + τ hτ τ )/τ }K We also introduce the piecewise linear interpolant of the values {(hkτ − hk−1 τ k=1 (namely, the values taken 0 by the -piecewise constant- function hτ ), viz.

b hτ : (0, T ) → X

t − tτk−1 hkτ − hk−1 tk − t hk−1 − hk−2 τ τ τ b hτ (t) := + τ τ τ τ τ

for t ∈ (tk−1 , tkτ ]. τ

Note that b h0τ (t) = D2τ,k (h) for t ∈ (tk−1 , tkτ ]. τ Furthermore, we denote by tτ and by tτ the left-continuous and right-continuous piecewise constant interpolants associated with the partition, i.e. tτ (t) := tkτ if tk−1 < t ≤ tkτ and tτ (t) := tk−1 if tk−1 ≤ t < tkτ . τ τ τ Clearly, for every t ∈ [0, T ] we have tτ (t) ↓ t and tτ (t) ↑ t as τ → 0. τ In view of (2.21), (2.22), and (2.23), it is easy to check that the piecewise constant interpolants (f τ )K k=1 , Kτ k k k τ (g τ )K k=1 , (hτ )k=1 of the values fτ , gτ , hτ (4.1) fulfill as τ ↓ 0

f τ → f in L2 (0, T ; L2 (Ω; Rd )), 1

1

(4.47)

2

0

1

g τ → g in L (0, T ; L (Ω)) ∩ L (0, T ; H (Ω) ).

(4.48)

hτ → h in L1 (0, T ; L2 (∂Ω)).

(4.49)

b τ , χτ , We now rewrite the discrete equations (4.4)–(4.6) in terms of the interpolants ϑτ , ϑτ , uτ , uτ , uτ , u τ χτ , χτ , ξ τ , and ζ τ of the elements (ϑkτ , ukτ , χkτ , ξτk , ζτk )K k=1 . Indeed, we have for almost all t ∈ (0, T ) ∂t ϑτ (t) + ∂t χτ (t)ϑτ (t) + ρ div(∂t uτ (t))ϑτ (t) + A

¯ tτ (t) τ

(ϑτ (t)) = g τ (t)+ 2

+ a(χτ (t))ε (∂t uτ (t)) Vε (∂t uτ (t)) + (1 + τ 1/2 ) |∂t χτ (t)| b τ (t) + V (a(χτ (t))∂t uτ (t)) + E b(χτ (t))uτ (t) + Cρ (ϑτ ) = f τ (t) ∂t u

in X ∗ ,

a.e. in Ω, (1 +

√

(4.50)

(4.51)

τ )∂t χτ (t) + µζ τ (t) + Ap χτ (t) + ξ τ (t)+γ(χτ (t)) ε(uτ (t))Eε(uτ (t)) + ϑτ (t) = −b0 (χτ (t)) 2 a.e. in Ω,

(4.52)

with ξ τ ∈ β(χτ ) and ζ τ ∈ ∂I(−∞,0] (∂t χτ ) a.e. in Ω × (0, T ). Our next result states that the interpolants of suitable discrete solutions to system (4.4)–(4.6) also satisfy the approximate versions of the entropy inequality (2.37) and of the total enegy inequality (2.38). For stating the discrete entropy inequality (4.55) below, we need to introduce discrete test functions. Namely, with every test function ϕ ∈ C0 ([0, T ]; W 1,d+ (Ω)) ∩ H 1 (0, T ; L6/5 (Ω)) we associate for k = 1, . . . , Kτ

ϕkτ := ϕ(tkτ )

(4.53)

τ and consider the piecewise constant and linear interpolants ϕτ and ϕτ of the values (ϕkτ )K k=1 . It can be shown that the following convergences hold as τ → 0

ϕτ → ϕ

in L∞ (0, T ; W 1,d+ (Ω)) and

∂t ϕτ → ∂t ϕ

Then, (4.55) is obtained by testing (4.4) by ϕkτ /ϑkτ , for k = 1, . . . , Kτ .

in L2 (0, T ; L6/5 (Ω)).

(4.54)

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As for the total energy inequality (4.56) below, let us mention that it results from our carefully designed time-discretization scheme, observing in addition that (4.6) is indeed the Euler-Lagrange equation for a suitable minimum problem, cf. (4.58) below, where the additional term Z χk χk−1 2 τ− τ τ 3/2 dx 2 Ω τ R has the role to “compensate” for the possible non-convexity of Ω γ b(χ) dx. Accordingly, to get the discrete 2 1/2 χk −χk−1 total energy inequality (4.56) we need to add the term τ 2 τ τ τ to the right-hand side of (4.4). This will lead to the necessary cancellations, cf. (4.66) below. Proposition 4.8 (Discrete entropy and total energy inequalities, µ ∈ {0, 1}). Under Hypotheses (I)–(III), τ for τ > 0 sufficiently small, the discrete solutions (ϑkτ , ukτ , χkτ )K k=1 to Problem 4.2 fulfill - the discrete entropy inequality Z tτ (t) Z Z (log(ϑτ (r)) + χτ (r))∂t ϕτ (r+τ ) dx dr + ρ tτ (s)

Ω

Z

tτ (t)

Ω

Z K(ϑτ (r))∇ log(ϑτ (r)) · ∇ϕτ (r) dx dr

tτ (s)

Z

Z div(∂t uτ (r))ϕτ (r) dx dr

tτ (s)

− Ω

(log(ϑτ (tτ (t))) + χτ (tτ (t)))ϕ(tτ (t)) dx −

≤

tτ (t)

Ω

Z

(log(ϑτ (tτ (s))) + χτ (tτ (s)))ϕ(tτ (s)) dx

Ω

Z

tτ (t)

Z

−

K(ϑτ (r)) tτ (s)

Z

tτ (t)

Ω

Z

− tτ (s)

Z

tτ (t)

g τ (r) + a(χτ (r))ε(∂t uτ (r))Vε(∂t uτ (r)) + |∂t χτ (r)|2 +

Ω

Z

−

hτ (r) tτ (s)

(4.55)

ϕτ (r) ∇ log(ϑτ (r)) · ∇ϑτ (r) dx dr ϑτ (r)

∂Ω

τ 3/2 χ |∂t τ (r)|2 2

ϕτ (r) dx dr ϑτ (r)

ϕτ (r) dS dr ϑτ (r)

for all 0 ≤ s ≤ t ≤ T and for all ϕ ∈ C0 ([0, T ]; W 1,d+ (Ω)) ∩ H 1 (0, T ; L6/5 (Ω)) with ϕ ≥ 0; - the discrete total energy inequality for all 0 ≤ s ≤ t ≤ T , viz. Z tτ (t) Z E (ϑτ (t), uτ (t), ∂t uτ (t), χτ (t)) ≤ E (ϑτ (s), uτ (s), ∂t uτ (s), χτ (s)) + (g τ + f τ · ∂t uτ ) dx dr tτ (s)

Z

Ω

tτ (t)

(4.56)

Z

+

hτ dS dr , tτ (s)

∂Ω

with E from (2.39). For the proof of the discrete entropy inequality, we will rely on a crucial inequality satisfied by any concave function ψ : dom(ψ) → R, i.e. ψ(x) − ψ(y) ≤ ψ 0 (y)(x − y)

for all x, y ∈ dom(ψ).

Proof. We split the proof in two steps. Step 1: proof of the total energy inequality. Let us consider the minimum problem n Z τ 3/2 χ − χk−1 2 χk − χk−1 χ − χk−1 |∇χ|p τ τ τ τ b χ) + χ + µb min α + + β( χ∈W 1,p (Ω) 2 τ τ τ p Ω o ε(uk−1 )Eε(uk−1 ) τ τ − ϑkτ χ dx +γ b(χ) + b(χ) 2

(4.57)

(4.58)

30

ELISABETTA ROCCA AND RICCARDA ROSSI

where χkτ is the discrete solution from Lemma 4.4, and let λ > 0 such that γ b00 ≥ −λ as in (2.19). Then, the function r 7→ γ b(r) + λ|r|2 is strictly convex. (4.59) Let τ¯ > 0 such that 1/(2τ ) > λ for all 0 < τ ≤ τ¯. We may rewrite the minimum problem (4.58) as k nZ 1 χ − χk−1 χτ − χk−1 |∇χ|p τ τ k−1 2 b χ) + γ χ χ χ √ −λ | − τ | + min + µb α + + β( b(χ) χ∈W 1,p (Ω) τ τ p 2 τ Ω o ε(uk−1 )Eε(uk−1 ) τ τ + λ|χ|2 + b(χ) − ϑkτ χ + λ|χk−1 |2 + 2λχχk−1 dx . τ τ 2 (4.60) b Observe that the Euler-Lagrange equation for (4.60) is exactly (4.6). Using the convexity of α b, β, b, and the λ-convexity of γ b (whence (4.59)), it is not difficult to check that (4.6) has a unique solution. We may thus conclude that the minimum problem (4.60) has a unique solution, which coincides with the discrete solution χkτ from Lemma 4.4. Now, choosing χk−1 as a competitor for χkτ in the minimum problem (4.58) yields τ k Z 3/2 χk χk−1 2 Z Z Z Z χk χk−1 2 χτ − χk−1 τ− τ τ− τ τ |∇χkτ |p τ b χk ) dx dx + dx + µ α b τ dx + dx + β( τ τ τ τ p Ω 2 Ω Ω Ω Ω Z Z Z ε(uk−1 )Eε(uk−1 ) τ τ + γ b(χkτ ) dx + b(χkτ ) dx − ϑkτ χkτ dx 2 Ω Ω Ω Z Z Z |p |∇χk−1 τ b χk−1 ) dx + dx + β( γ b(χk−1 ) dx ≤ τ τ p Ω Ω Ω Z Z k−1 ) )Eε(uk−1 τ k−1 ε(uτ χ + b( τ ) dx − ϑkτ χk−1 dx. τ 2 Ω Ω (4.61) and observe that , for all k = 1, . . . , K , Hence, we test (4.5) by ukτ − uk−1 τ τ Z 1 1 τ D2τ,k (u) · Dτ,k (u) dx ≥ kDτ,k (u)k2L2 (Ω) − kDτ,k−1 (u)k2L2 (Ω;Rd ) (4.62) 2 2 Ω Z k ukτ − uk−1 ukτ − uk−1 τ τ k−1 k−1 χ Vε dx. (4.63) hV a(χk−1 )D (u) , u − u i = τ a( )ε τ,k 1 τ τ τ τ H (Ω) τ τ Ω Furthermore, we have Z Z uk − uk−1 1 1 τ i ≥ b(χkτ )ε(ukτ )Eε(ukτ ) dx − b(χkτ )ε(uk−1 )Eε(uk−1 ) dx hE b(χkτ )ukτ , τ τ τ τ 2 Ω 2 Ω H 1 (Ω;Rd ) Z Z 1 1 = b(χkτ )ε(ukτ )Eε(ukτ ) dx − b(χk−1 )ε(uk−1 )Eε(uk−1 ) dx τ τ τ 2 Ω 2 Ω Z 1 − (b(χkτ ) − b(χk−1 ))ε(uk−1 )Eε(uk−1 ) dx . τ τ τ 2 Ω

(4.64)

Finally, k Z ukτ − uk−1 uτ − uk−1 τ τ k τ i = −ρ ϑτ div dx . (4.65) τ τ H 1 (Ω;Rd ) Ω Next, we multiply (4.4) by τ and integrate over Ω. We add the resulting relation to the equation obtained testing (4.16) by ukτ − uk−1 and to (4.61). The terms τ Z Z k χ τ Dτ,k ( )ϑτ dx, ρτ ϑkτ div(Dτ,k (u)) dx, Ω Ω k k Z χk χk−1 2 Z τ− τ uτ − uk−1 uτ − uk−1 τ τ dx τ a(χk−1 )ε Vε dx, τ (4.66) τ τ τ τ Ω Ω Z χk χk−1 2 Z τ− τ τ 3/2 1 dx, (b(χkτ ) − b(χk−1 ))ε(uk−1 )Eε(uk−1 ) dx τ τ τ 2 Ω τ 2 Ω hCρ (ϑkτ ),

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

31

cancel out. We sum over the index k = m, . . . , j, for any couple of indexes 1 ≤ m < j ≤ Kτ . Taking into account (4.61)–(4.65), we ultimately obtain Z 1 1 |∇χjτ |p j b χj ) + γ χ ϑjτ + |Dτ,j (u)|2 + b( χjτ )ε(ujτ )Eε(ujτ ) + + β( b ( ) dx τ τ 2 2 p Ω Z p 1 χm 1 |∇χm τ | 2 m m m m b χ χ ≤ |D (u)| + b( )ε(u + β( ) + γ b ( ) dx ϑm + )Eε(u ) + τ,m τ τ τ τ τ τ (4.67) 2 2 p Ω Z Z j X gτk + fτk · Dτ,k (u) dx + + τ hkτ dS , k=m

Ω

∂Ω

which yields (4.56). Step 2: proof of the entropy inequality. Let us fix an arbitrary posititive test function ϕ ∈ C0 ([0, T ]; W 1,d+ (Ω)) ∩ H 1 (0, T ; L6/5 (Ω)) ϕk

τ τ ∈ H 1 (Ω) (hence, an admissible test function for (4.4)) with (ϕkτ )K k=1 defined by (4.53). We multiply (4.4) by ϑk τ and integrate over Ω. We obtain 2 2 ! k k k k Z k−1 k−1 1/2 χk χk−1 χτ − χk−1 − u − u τ − ϕτ u u τ τ τ τ τ τ τ k k + dx gτ + a(χτ )ε Vε + τ τ τ 2 τ ϑkτ Ω Z ϕk + hkτ kτ dS ϑτ ∂Ω k k Z k Z χk − χk−1 ϑτ − ϑk−1 uτ − uk−1 ϕτ τ τ τ = + τ ϑkτ + ρ div ϑkτ ϕkτ dx + K(ϑkτ )∇ϑkτ · ∇ dx (4.68) τ τ τ ϑkτ Ω Ω k Z uτ − uk−1 log(ϑkτ ) − log(ϑk−1 ) χkτ − χk−1 τ τ τ + + ρ div ϕkτ dx ≤ τ τ τ Ω Z K(ϑkτ ) K(ϑkτ ) k k k 2 k + ∇ϑτ · ∇ϕτ − k 2 |∇ϑτ | ϕτ dx ϑkτ |ϑτ | Ω

where we have used that (cf. (4.57)) ϑkτ − ϑk−1 τ ≤ log(ϑkτ ) − log(ϑk−1 ) τ ϑkτ

a.e. in Ω.

Note that this inequality is preserved by the positivity of the discrete test function ϕkτ . We now sum (4.68), multiplied by τ , over k = m, . . . , j, for any couple of indexes 1 ≤ m < j ≤ Kτ . We use the discrete integration by parts formula (4.46), yielding j X

Z τ Ω

k=m j X k=m

Dτ,k (log(ϑkτ ))ϕkτ dx =

Z τ Ω

Z

log(ϑjτ )ϕjτ dx −

Ω

Dτ,k (χkτ )ϕkτ dx =

Z Ω

χjτ ϕjτ dx −

Z

m+1 log(ϑm dx − τ )ϕτ

Ω

Z Ω

m+1 χm dx − τ ϕτ

j−1 X k=m

j−1 X k=m

Z τ

Z τ

log(ϑkτ )Dτ,k+1 (ϕ) dx

Ω

χkτ Dτ,k+1 (ϕ) dx.

Ω

Inserting the two above inequalities in (4.68) (summed up over k = m, . . . , j), rearranging terms, we conclude (4.55). Remark 4.9. A close perusal of the proof of Proposition 4.8 reveals that, b is only λ-convex, in place of convex, it is still possible to prove that the discrete equation for χ (4.6) admits a unique solution, and therefore conclude that χkτ is the unique minimizer for (4.58). This, provided we replace the p-Laplacian operator in (4.6) with its non-degenerate version, cf. Remark 2.9.

32

ELISABETTA ROCCA AND RICCARDA ROSSI

4.3. A priori estimates revisited. The following result collects all the a priori estimates for the approximate solutions constructed via time discretization. In particular, the proof renders on the discrete level the Second and Sixth estimates, which have a nonlinear character and thus translate with some difficulty within the frame of the discrete system (4.4)–(4.6). In particular, the Sixth estimate (cf. (4.69h) below) is deduced with careful calculations from the discrete entropy inequality (4.55). Proposition 4.10. Assume Hypotheses (I)–(III) and (2.21)–(2.26). Let µ ∈ {0, 1}. Then, there exists a constant S > 0 such that for all τ > 0 the following estimates kuτ kL∞ (0,T ;H02 (Ω;Rd )) ≤ S,

(4.69a)

kuτ kH 1 (0,T ;H02 (Ω;Rd ))∩W 1,∞ (0,T ;H01 (Ω;Rd )) ≤ S,

(4.69b)

kb uτ kH 1 (0,T ;L2 (Ω;Rd )) ≤ S,

(4.69c)

kχτ kL∞ (0,T ;W 1,p (Ω)) ≤ S, kχτ kL∞ (0,T ;W 1,p (Ω))∩H 1 (0,T ;L2 (Ω)) ≤ S,

(4.69d)

k log(ϑτ )kL2 (0,T ;H 1 (Ω)) ≤ S,

(4.69f)

kϑτ kL2 (0,T ;H 1 (Ω))∩L∞ (0,T ;L1 (Ω)) ≤ S,

(4.69g)

k log(ϑτ )kBV([0,T ];W 1,d+ε (Ω)∗ ) ≤ S

(4.69h)

(4.69e)

for all > 0

hold. Furthermore, under Hypothesis (V) (i.e. if 1 < κ < 5/3), we have in addition sup kϑτ kBV([0,T ];W 2,d+ (Ω)∗ ) ≤ S

for all > 0.

(4.69i)

τ >0

Finally, if µ = 0 we also have sup kχτ kL2 (0,T ;W 1+σ,p (Ω)) + kξ τ kL2 (0,T ;L2 (Ω)) ≤ S

for all 1 ≤ σ
0

1 . p

(4.69j)

We now sketch the proof, showing how the formal a priori estimates in Section 3 can be translated in the framework of the time discretization scheme; we shall only detail the argument for the discrete version of the Sixth estimate. Proof. From the discrete total energy inequality (4.56), arguing in the very same way as for the First a priori estimate, we deduce kϑτ kL∞ (0,T ;L1 (Ω)) + kuτ kW 1,∞ (0,T ;L2 (Ω;Rd )) + kχτ kL∞ (0,T ;W 1,p (Ω)) ≤ C,

(4.70)

whence (4.69d). We also infer that kb(χτ )1/2 ε(uτ )kL∞ (0,T ;L2 (Ω;Rd×d )) ≤ C which gives, via (2.16) and Korn’s inequality, that kuτ kL∞ (0,T ;H01 (Ω;Rd )) ≤ C. Next, along the lines of the Second a priori estimate, we test (4.4) by F 0 (ϑkτ ) = (ϑkτ )α−1 , with α ∈ (0, 1). Since F (ϑ) = ϑα /α is concave, by (4.57) we have (ϑkτ − ϑk−1 )F 0 (ϑkτ ) ≤ F (ϑkτ ) − F (ϑk−1 ) τ τ

a.e. in Ω,

therefore we obtain 2 ! k k Z Z uτ − uk−1 uτ − uτk−1 τ 1/2 χkτ − χk−1 τ τ k k 0 k χ gτ + a( τ )ε Vε + 1+ hkτ F 0 (ϑkτ ) dS F (ϑτ ) dx + τ τ 2 τ Ω ∂Ω k Z F (ϑkτ ) − F (ϑk−1 ) χkτ − χk−1 uτ − uk−1 τ τ τ k 0 k k 0 k k k 0 k ≤ + ϑτ F (ϑτ ) + ρ div ϑτ F (ϑτ ) + K(ϑτ )∇ϑτ ∇(F (ϑτ )) dx . τ τ τ Ω (4.71)

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

33

Then, we multiply (4.71) by τ . Summing over the index k and recalling that g ≥ 0 and h ≥ 0, we obtain for all t ∈ (0, T ] Z Z 4(1 − α) tτ (t) K(ϑτ )|∇((ϑτ )α/2 )|2 dx ds α2 0 Ω Z tτ (t) Z τ 1/2 2 0 2 0 χ c2 |ε(∂t uτ )| F (ϑτ ) + 1 + + |∂t τ | F (ϑτ ) dx ds 2 0 Ω Z Z Z tτ (t) Z ≤ F (ϑτ (t)) dx − F (ϑ0 ) dx + ∂t χτ ϑτ F 0 (ϑτ ) + ρ div(∂t uτ )ϑτ F 0 (ϑτ ) dx ds . Ω

Ω

0

Ω

Starting from this inequality, we develop calculations completely analogous to the ones in Section 3 for the Second a priori estimate. In particular, we conclude that Z tτ (t) Z K(ϑτ )|∇((ϑτ )α/2 )|2 dx ds ≤ C . (4.72) 0

Ω

The same calculations as for the Third estimate allow us then to deduce from (4.72) and (4.70) estimate (4.69g). As a byproduct of these calculations, we again have for all α ∈ [1/2, 1) k(ϑτ )(κ−α)/2 kL2 (0,T ;H 1 (Ω)) , k(ϑτ )(κ+α)/2 kL2 (0,T ;H 1 (Ω)) ≤ C .

(4.73)

Moreover, since ϑτ (t) ≥ ϑ

a.e. in Ω

for all t ∈ [0, T ],

(4.74)

(with ϑ from (4.25)), we also have (4.69f). As for the Fourth estimate, we subtract from the discrete total energy inequality (4.56) the discrete heat equation (4.4) multiplied by τ and summed over the index k. Therefore, we obtain for all t ∈ [0, T ] Z Z tτ (t) 1 1 2 v(a(χτ )∂t uτ , ∂t uτ ) ds + e(b(χτ (tτ (t)))uτ (tτ (t)), uτ (tτ (t))) |∂t uτ (tτ (t))| dx + 2 Ω 2 0 Z tτ (t) Z Z τ 1/2 1 χ + 1+ |∇ τ (tτ (t))|p + W (χτ (tτ (t))) dx |∂t χτ |2 dx ds + 2 0 Ω Ω p Z tτ (t) Z Z tτ (t) Z χ = I0 + ϑτ (ρ div(∂t uτ ) + ∂t τ ) dx ds + f τ · ∂t uτ dx ds , 0

Ω

0

Ω

+ p1 |∇χ0 |p + W (χ0 )) dx. Exploiting RtR (2.21) and estimate (4.69g), we control the second term on the right-hand side with 0 Ω |∂t χτ |2 dx ds and R t (t) the second term on the left-hand side, which bounds 0 τ k∂t uτ k2H 1 (Ω;Rd ) ds thanks to (2.5). Therefore, we conclude that k∂t uτ kL2 (0,T ;H 1 (Ω;Rd )) ≤ C, as well as estimate (4.69e). The Fifth estimate is performed on the time-discretization scheme by testing (4.5) by − div(ε(ukτ −uk−1 )). τ For all the calculations, we refer to [28, (3.61)–(3.67)]: therein, the equation for u was the same as our own (1.2), but the elasticity and viscosity tensors E and V were assumed to be independent of the space variable x. Nonetheless, the computations from [28] carry over to the present setting, cf. also the formal calculations for the Fourth a priori estimate in Sec. 3. Therefore, we conclude estimates (4.69a) and (4.69b). A comparison argument in (4.5), joint with (2.8b), yields (4.69c). In order to render the Sixth estimate in the time discrete setting, let us fix a partition 0 = σ0 < σ1 < . . . < σJ = T of the interval [0, T ]. Preliminarily, from the discrete entropy inequality (4.55), written on the interval [σi−1 , σi ] and for a constant-in-time test function ϕ ∈ W 1,d+ (Ω) for some > 0, we deduce that Z (`i − `i−1 )ϕ dx + Λi (ϕ) ≥ 0 for all ϕ ∈ W+1,d+ (Ω), (4.75) Ω Z (`i−1 − `i )ϕ dx − Λi (ϕ) ≥ 0 for all ϕ ∈ W−1,d+ (Ω), (4.76) where we have used the place-holder I0 =

Ω

1 χ 2 e(b( 0 )u0 , u0 )

+

R

( 1 |v |2 Ω 2 0

34

ELISABETTA ROCCA AND RICCARDA ROSSI

where we have used the place-holders `i = log(ϑτ (σi )) + χτ (σi ), Z tτ (σi ) Z Z K(ϑτ )∇ log(ϑτ ) · ∇ϕ dx dr − ρ Λi (ϕ) = tτ (σi−1 )

Ω

tτ (σi )

Z div(∂t uτ )ϕ dx dr

tτ (σi−1 )

Z tτ (σi ) Z ϕ ϕ − K(ϑτ ) ∇(log(ϑτ ))∇ϑτ dx dr − hτ dS dr ϑτ ϑτ tτ (σi−1 ) Ω tτ (σi−1 ) ∂Ω Z tτ (σi ) Z τ 1/2 ϕ 2 χ χ − g τ + a( τ )ε(∂t uτ )Vε(∂t uτ ) + 1 + |∂t τ | dx dr. 2 ϑτ tτ (σi−1 ) Ω Z

tτ (σi )

Ω

Z

For later use, we also introduce the place-holder τ 1/2 1 |∂t χτ |2 , Rτ := ρ div(∂t uτ ) + K(ϑτ )|∇(log(ϑτ ))|2 + g τ + a(χτ )ε(∂t uτ )Vε(∂t uτ ) + 1 + 2 ϑτ such that Z

tτ (σi )

Z

Λi (ϕ) = tτ (σi−1 )

K(ϑτ )∇ log(ϑτ ) · ∇ϕ − Rτ ϕ dx dr −

Z

tτ (σi )

Z hτ

Ω

tτ (σi−1 )

∂Ω

ϕ dS dr . ϑτ

(4.77)

We also deduce from (4.75) with ϕ ≡ 1 that Z

tτ (σi )

Z

tτ (σi−1 )

Ω

`i − `i−1 − Rτ tτ (σi ) − tτ (σi−1 )

Z

tτ (σi )

Z

dx dr ≥

hτ tτ (σi−1 )

∂Ω

1 dS dr ≥ 0. ϑτ

(4.78)

We now estimate the total variation of (log(ϑτ ) + χτ ) with values in W 1,d+ (Ω)∗ , i.e. VarW 1,d+ (Ω)∗ (log(ϑτ ) + χτ ; [0, T ]) =

J X

sup

k(log(ϑτ (σi ) + χτ (σi )) − (log(ϑτ (σi−1 ) + χτ (σi−1 ))kW 1,d+ (Ω)∗

0=σ0 0 and for all t ∈ [0, T ],

(5.10)

log(ϑτk (t)) * log(ϑ(t))

in W

(5.11)

ϑτ k → ϑ

in Lh (Ω × (0, T )) for all h ∈ [1, 8/3) for d = 3 and all h ∈ [1, 3) if d = 2,

(5.12)

log ϑ ∈ BV([0, T ]; W 1,d+ (Ω)∗ ) for all > 0,

(5.13)

and ϑ also fulfills ϑ ∈ L∞ (0, T ; L1 (Ω)),

ϑ ≥ ϑ a.e. in Ω × (0, T )

(with ϑ from (4.10)). Under the additional Hypothesis (V), we also have ϑ ∈ BV([0, T ]; W 2,d+ (Ω)∗ ) for all > 0, and ϑτ k → ϑ ϑτk (t) → ϑ(t)

in L2 (0, T ; Y ) for all Y such that H 1 (Ω) b Y ⊂ W 2,d+ (Ω)∗ , in W

2,d+

∗

(Ω) for all t ∈ [0, T ].

(5.14) (5.15)

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

37

Proof. Due to due to estimates (4.69b) and (4.69c), there holds kuτ − uτ kL∞ (0,T ;H02 (Ω;Rd )) ≤ τ 1/2 k∂t uτ kL2 (0,T ;H02 (Ω;Rd )) ≤ Sτ 1/2 , b τ kL2 (0,T ;L2 (Ω;Rd )) ≤ Sτ 1/2 . kb uτ − ∂t uτ kL∞ (0,T ;L2 (Ω;Rd )) ≤ τ 1/2 k∂t u

(5.16)

Taking into account (4.69a), (4.69b), (4.69c), and applying well-known weak and strong compactness results (for the latter, cf. e.g. [32]), we conclude convergences (5.1)–(5.5). The same kind of arguments yields (5.6)– (5.7) on account of estimates (4.69d) and (4.69e). The bound (4.69g) gives the weak convergence (5.8). Since the family (log(ϑτ ))τ is bounded in L2 (0, T ; H 1 (Ω)) ∩ BV([0, T ]; W 1,d+ (Ω)∗ ) for all > 0, an Aubin-Lions ˙ or [29, Chap. 7, Cor. 4.9]) ensures type compactness result for BV-functions (see, for instance, [32, Cor4] 2 that, up to a subsequence, log(ϑτk ) converges to some λ in L (0, T ; Z) for every Banach space Z such that H 1 (Ω) b Z ⊂ W 1,d+ (Ω)∗ . Therefore, log(ϑτk ) converges to λ pointwise almost everywhere in Ω × (0, T ) and accordingly ϑτk converges to eλ . Then, in view of (5.8), λ = log(ϑ), and convergences (5.9) and (5.10) ensue. The BV-compactness result [22, Thm. 6.1] also ensures that log(ϑ) ∈ BV([0, T ]; W 1,d+ (Ω)∗ ), and the additional weak convergence (5.11). With a lower semicontinuity argument one also has that ϑ ∈ L∞ (0, T ; L1 (Ω)), and convergence (5.12) follows from an interpolation argument (cf. (3.15)). Relying on this and on the approximate positivity property (4.74), we also conclude the last of (5.13). Under the additional Hypothesis (V), we also dispose of the BV-estimate (4.69i) for ϑτ . Combining this with (4.69g) and applying the aforementioned compactness results from [32] and [22], we conclude (5.14)–(5.15). We are now in the position to develop the Proof of Theorem 2.5, by passing to the limit in the time-discrete scheme set up in Sec. 4. Let (τk )k be a vanishing sequence of time-steps, and let b τk , χτk , χτk , χτk )k (ϑτk , ϑτk , uτk , uτk , uτk , u be a sequence of approximate solutions. We can exploit the compactness results from Lemma 5.1. We split the limit passage in the following steps. Ad the weak momentum equation (2.40). Relying on convergences (5.1), (5.4)–(5.5), (5.7) and (5.8), as well as (4.47) for (f τk )k , we pass to the limit in (4.51) and conclude that the triple (ϑ, u, χ) fulfills (2.40). Ad the weak formulation (2.41)–(2.44) of the equation for χ, µ = 1. The argument for obtaining (2.41)– (2.44) in the limit follows exactly the same lines as the proof of [16, Thms. 4.4, 4.6] (see also [28, Thm. 3]). Therefore we only recapitulate it, referring to the latter papers for all details. First of all, as we have pointed out in the proof of Proposition 4.8, (4.6) can be interpreted as the EulerLagrange equation for the minimum problem (4.58), i.e. (recall that here µ = 1 and that α b = I(−∞,0] and βb = I[0,+∞) ) n Z τ 3/2 min χ∈W 1,p (Ω) 2 Ω

2 k χτ − χk−1 χ − χk−1 χ − χk−1 |∇χ|p τ τ τ + χ + I(−∞,0] + + I[0,+∞) (χ) τ τ τ p o ε(uk−1 )Eε(uk−1 ) τ τ +γ b(χ) + b(χ) − ϑkτ χ dx 2

(5.17)

Writing necessary optimality conditions for the minimum problem (5.17), with the very same calculations as for [28, Thm. 3], we arrive at Z √ ∂t χτ (t)ψ + τ ∂t χτ (t)ψ + |∇χτ (t)|p−2 ∇χτ (t) · ∇ψ + γ(χτ (t))ψ + j τ (t)ψ dx ≥ 0 (5.18) Ω for all t ∈ [0, T ] and all ψ ∈ W 1,p (Ω) s.t. there exists ν > 0 with 0 ≤ νψ + χτ (t) ≤ χτ (t) a.e. in Ω, where where we have used the place-holder j τ := b0 (χτ )

ε(uτ )Eε(uτ ) − ϑτ . 2

(5.19)

38

ELISABETTA ROCCA AND RICCARDA ROSSI

Choosing ψ = −∂t χτ (t) in (5.18) and and summing over the index k we deduce the discrete version of the energy inequality (2.44) for all 0 ≤ s ≤ t ≤ T , viz. tτ (t)

1 χ p χ (1 + τ |∇ τ (tτ (t)))| + W ( τ (tτ (t))) dx p tτ (s) Ω Ω Z 1 χ ≤ |∇ τ (tτ (s))|p + W (χτ (tτ (s))) dx p Ω Z tτ (t) Z 0 χ ε(uτ )Eε(uτ ) χ + ∂t τ −b ( τ ) + ϑτ dx dr + Cτ k∂t χτ k2L2 (0,T ;L2 (Ω)) , 2 tτ (s) Ω

Z

Z

1/2

)|∂t χτ |2 dx dr +

where we have used that Z tτ (t) Z χ χ γ( τ )∂t τ dx dr = tτ (s)

Z

tτ (t)

γ(χτ )∂t χτ dx dr +

tτ (s)

Z

tτ (t)

(5.20)

. γ(χτ ) − γ(χτ ) ∂t χτ dx dr = I1 + I2

tτ (s)

and that, by the chain rule, Z Z Z Z χ χ χ γ b( τ (tτ (s))) dx = W ( τ (tτ (t))) dx − W (χτ tτ (s))) dx I1 = γ b( τ (tτ (t))) dx − Ω

Ω

Ω

Ω

(due to βb = I[0,+∞) ), while I2 ≤ k∂t χτ kL2 (0,T ;L2 (Ω)) kγ(χτ ) − γ(χτ )kL2 (0,T ;L2 (Ω)) ≤ Cτ k∂t χτ k2L2 (0,T ;L2 (Ω)) thanks to the Lipschitz continuity of γ. Second, repeating the “recovery sequence” argument from [16, proof of Thm. 4.4], we improve the weak convergence (5.6) to χτk → χ

in Lp (0, T ; W 1,p (Ω)).

(5.21)

We refer to [16] and [28] for all the related calculations. We are now in the position to taking the limit as τk ↓ 0 in the approximate energy inequality (5.20). We pass to the limit on the left-hand side by lower semicontinuity, relying on convergences (5.6) and (5.21). For the right-hand side, we exploit the latter strong convergence as well as (5.7), yielding that χτk (s) → χ(s) b ∈ C2 (R) that γ b in W 1,p (Ω), whence χτk (s) → χ(s) in C0 (Ω), for almost all s ∈ (0, T ). It follows from γ has at most quadratic growth on bounded subsets of R. We combine this with the uniform convergence of R R (χτk (s))k to conclude that Ω γ b(χτk (s)) dx → Ω γ b(χ(s)) dx for almost all s ∈ (0, T ). Since βb = I[0,+∞) , we R R χ χ have Ω W ( τk (s)) dx → Ω W ( (s)) dx for almost all s ∈ (0, T ). Since (χτ )τ is bounded in H 1 (0, T ; L2 (Ω)), we also have √ τ ∂t χτk → 0 in L2 (0, T ; L2 (Ω)). (5.22) Combining the weak convergence (5.6) with the strong ones (5.2), (5.7), and (5.12), we also pass to the limit in the second integral term on the right-hand side of (5.20). The last summand obviously tends to zero. Therefore, we conclude the energy inequality (2.44). Clearly, convergence (5.6) and the fact that ∂t χτ ≤ 0 a.e. in Ω × (0, T ) ensure that χt ≤ 0 .e. in Ω × (0, T ), i.e. (2.41). To obtain the variational inequality (2.42), together with (2.43), we proceed exaclty as in [16, 28]. The main steps are as follows: passing to the limit in (5.18) as τk ↓ 0 with suitable test functions from [16, Lemma 5.2], also relying on (5.22), we prove that for almost all t ∈ (0, T ) Z χt (t)ψ˜ + |∇χ(t))|p−2 ∇χ(t) · ∇ψ˜ + γ(χ(t))ϕ˜ + b0 (χ(t)) ε(u(t))Eε(u(t)) ψ˜ − ϑ(t)ψ˜ dx ≥ 0 2 Ω 1,p for all ψ˜ ∈ W− (Ω) with {ψ˜ = 0} ⊃ {χ(t) = 0}.

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

From this, arguing as in the proof of [16, Thm. 4.4] we deduce that for almost all t ∈ (0, T ) Z χt (t)ψ + |∇χ(t))|p−2 ∇χ(t) · ∇ψ + γ(χ(t))ϕ + b0 (χ(t)) ε(u(t))Eε(u(t)) ψ − ϑ(t)ψ dx 2 Ω + Z ε(u(t))Eε(u(t)) ψ dx for all ψ ∈ W−1,p (Ω). ≥ − ϑ(t) γ(χ(t)) + b0 (χ(t)) 2 {χ(t)=0}

39

(5.23)

Relying on (5.23), it is possible to check that the function ξ from (2.47) complies with (2.42) and (2.43). Ad the entropy inequality (2.37). Let us fix a test function ϕ ∈ C0 ([0, T ]; W 1,d+ (Ω)) ∩ H 1 (0, T ; L6/5 (Ω)) (for some > 0), for the entropy inequality (2.37). We pass to the limit as τk ↓ 0 in the discrete entropy inequality (4.55), with the discrete test functions constructed from ϕ in (4.53). In order to pass to the limit in the first two integral terms on the left-hand side of (4.55), we combine convergences (5.1), (5.7), and (5.10), with the convergence (4.54) for the test functions. In order to deal with the last integral on the left-hand side, we observe that the family (K(ϑτ )∇ log(ϑτ ))τ is bounded in L1+δ (Q; Rd ) for some δ > 0. Indeed, the growth condition (2.15) implies that 1 1 κ−1 κ−1 |K(ϑτ )∇ log(ϑτ )| ≤ C |ϑτ | + + |∇ϑτ | |∇ϑτ | ≤ | ≤ C |ϑτ | ϑ(T ) ϑτ

(5.24)

a.e. in Ω × (0, T )

(also due to (4.74)). Thus, it remains to bound the term |ϑτ |κ−1 |∇ϑτ |. To do so, we observe ZZ r |ϑτ |κ−1 |∇ϑτ | dx dt ≤ k(|ϑτ |(κ−α)/2 )r kL2/(2−r) (Q) k(|ϑτ |(κ+α−2)/2 |∇ϑτk )r kL2/r (Q;Rd ) Q

(5.25)

≤ Ck(|ϑτ |(κ−α)/2 )r kL2/(2−r) (Q) for some r > 0 (to be chosen below), where we exploited that (|ϑτ |(κ+α−2)/2 ∇ϑτ )τ is bounded in L2 (Q; Rd ) thanks to (3.16) (cf. also (3.13)). Indeed the latter estimate yields that ((ϑτ )(κ+α)/2 )τ is bounded in L2 (Q), hence that ((ϑτ )(κ−α)/2 )τ is bounded in L2(κ+α)/(κ−α) (Q). Therefore, it is sufficient to choose in (5.25) r such that 2r/(2 − r) = 2(κ + α)/(κ − α), i.e. r = (κ + α)/κ, which is strictly bigger than 1. Therefore, up to some subsequence K(ϑτk )∇ log(ϑτk ) weakly converges to some η in L1+δ (Q; Rd ). In order to identify η as K(ϑ)∇ log(ϑ), we use these facts. We first show that |ϑτk |(κ+α−2)/2 ∇ϑτk * |ϑ|(κ+α−2)/2 ∇ϑ

in L2 (Q; Rd ).

(5.26)

Indeed, on the one hand, (5.8) gives ∇ϑτk * ∇ϑ in L2 (0, T ; L2 (Ω; Rd )). On the other hand, the pointwise convergence ϑτk → ϑ a.e. in Ω × (0, T ) combined with the fact that (ϑτk )k is bounded in Lκ+α (Ω) yields that ϑτk → ϑ in Lκ+α− (Ω) for all > 0. Therefore |ϑτ |(κ+α−2)/2 → |ϑ|(κ+α−2)/2 in L2(κ+α)/(κ+α−2)− (Ω) for all > 0 . Since (|ϑτk |(κ+α−2)/2 ∇ϑτk )k is bounded in L2 (Q; Rd ), (5.26) follows. Second, we have that |ϑτk |(κ−α)/2 → ϑ(κ−α)/2 in L2(κ+α)/(κ−α)− (Ω)

for all > 0,

(5.27)

again due to the pointwise convergence of ϑτk and to the fact (ϑτk )k is bounded in Lκ+α (Ω). It follows from (5.26), (5.27) , and the growth condition on K, that K(ϑτk )∇ log(ϑτk ) * K(ϑ)∇ log(ϑ)

in L1+δ (Q; Rd ).

(5.28)

This and convergence (4.54) enables us to take the limit in third term on the left-hand side of (4.55). The passage to the limit in the first two integrals on the right-hand side results from convergences (5.7), (5.11), and again (4.54). For the third term, we use that Z tZ Z tτ (t) Z 2 k 2 lim inf K(ϑτk (r))ϕτk (r) ∇ log(ϑτk (r)) dx dr ≥ K(ϑ(r))ϕ(r) |∇ log(ϑ(r))| dx dr k→∞

tτk (s)

Ω

s

Ω

40

ELISABETTA ROCCA AND RICCARDA ROSSI

which results from the weak convergence (5.9), combined with the pointwise convergence ϑτk → ϑ a.e. in Ω × (0, T ), (4.54) for the discrete test functions, applying the Ioffe theorem [18]. With analogous arguments we pass to the limit in the last two integrals on the right-hand side of (4.55), and therefore conclude (2.37). Ad the total energy inequality (2.38). It follows from passing to the limit as τk ↓ 0 in the discrete total energy inequality (4.56), based on convergences (4.47)–(4.49) for f τk , g τk , hτk , and on (5.2), (5.5), (5.7), and (5.12). Observe that convergences (5.2), (5.5), and (5.7) are sufficient to pass to the limit on the left-hand side of (4.56), by lower semicontinuity, for all t ∈ [0, T ]. However, (5.12) only guarantees that ϑτk (t) → ϑ(t) in L1 (Ω) for almost all t ∈ (0, T ). Enhanced regularity and improved total energy inequality under Hypothesis (V). If in addition Hyp. (V) holds, in view of Lemma 5.1 ϑ is in BV([0, T ]; W 2,d+ (Ω)∗ ) for every > 0, and the enhanced convergences (5.14) and (5.15) hold. The latter pointwise convergence allows us to pass to the limit on the left-hand side of (4.56) for all t ∈ [0, T ]. This concludes the proof. We conclude this section with the Proof of Theorem 2.8: Let (τk )k be a vanishing sequence of time-steps, b τk , χτk , χτk , χτk )k be a sequence of approximate solutions; let (ξ τk )k be a sequence and (ϑτk , ϑτk , uτk , uτk , uτk , u of selections in β(χτk ), such that (χτk , ξ τk ) satisfy for all k ∈ N the approximate equation (4.52). In the case µ = 0, in addition to convergences (5.1)–(5.15), estimates (4.69j) yield, up to a subsequence, the further convergences χτk * χ in L2 (0, T ; W 1+σ,p (Ω)) for all 1 ≤ σ < 1 , p

χτk * χ

in Lq (0, T ; W 1,p (Ω)) for all 1 ≤ q < ∞. (5.29)

Furthermore, there exists ξ ∈ L2 (0, T ; L2 (Ω)) such that ξ τk * ξ

in L2 (0, T ; L2 (Ω)).

(5.30)

The strong convergence (5.29) and the strong-weak closedness of β (as a maximal monotone operator from L2 (Ω) to L2 (Ω)) immediately yield that ξ ∈ β(χ) a.e. in Ω × (0, T ). Therefore, also exploiting convergences (5.1)–(5.8) we pass to the limit in the discrete equation for χ (4.52) and immediately conclude that the quadruple (ϑ, u, χ, ξ) fulfills the pointwise formulation (2.51)–(2.52) of the internal parameter equation (1.3). The proof of the entropy inequality, of the total energy inequality, and of the momentum equation is clearly the same as for Theorem 2.5. Under the additional Hypothesis (V), as previously seen ϑ is in BV([0, T ]; W 2,d+ε (Ω)∗ ). We prove the weak form (2.54) of the heat equation by passing to the limit as τk ↓ 0 in the approximate heat equation (4.50), tested by an arbtitrary ϕ ∈ C0 ([0, T ]; W 2,d+ (Ω)) ∩ H 1 (0, T ; L6/5 (Ω)). The passage to the limit in the first three terms on the left-hand side, and on the first two terms on the right-hand side, results from convergences (4.48), (4.49) for (g τk )k and (hτk )k , and from (5.1)–(5.2), (5.5)–(5.8): in particular, we exploit that ε(∂t uτk )Eε(∂t uτk ) → ε(ut )Eε(ut ) strongly in L1 (Q) thanks to the strong convergence (5.5). In order to pass to the limit with the fourth term on the left-hand side of (4.50), we need to derive a finer estimate for (K(ϑτk )∇ϑτk )k . Arguing as for (3.28) we use that |K(ϑτk )∇ϑτk | ≤ C|ϑτk |(κ−α+2)/2 |ϑτk |(κ+α−2)/2 |∇ϑτk | + C|∇ϑτk |. Now, (ϑτk )(κ+α−2)/2 ∇ϑτk is bounded in L2 (0, T ; L2 (Ω; Rd )) (thanks to (4.72)). On the other hand, (ϑτk )k is bounded in Lp (Q) for all 1 ≤ p < 8/3, in the case d = 3 (to which we confine this discussion). Therefore, choosing α ∈ [1/2, 1) such that α > κ − 32 (this can be done since κ < 5/3 by assumption), we conclude that ((ϑτk )(κ−α+2)/2 )k is bounded in L2+δ (Q) for some δ > 0. Ultimately, we conclude that (K(ϑτk )∇ϑτk )k is ¯ ¯ bounded in L1+δ (0, T ; L1+δ (Ω)) for some δ¯ > 0, hence ¯

¯

∃ η ∈ L1+δ (0, T ; L1+δ (Ω)) :

¯

¯

K(ϑτk )∇ϑτk * η in L1+δ (0, T ; L1+δ (Ω)) .

(5.31)

b τ ) a.e. in In order to identify the weak limit η, it is sufficient to observe that (cf. [21]) K(ϑτk )∇ϑτk = ∇K(ϑ k Ω × (0, T ). Combining the growth property (2.15) of K (where 1 ≤ κ < 5/3), with the strong convergence

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

41

b τ ))k strongly converges to K(ϑ) b (5.12) of ϑτk in Lp (Q) for all 1 ≤ p < 8/3, we ultimately conclude that (K(ϑ k 1+δ˜ ˜ in L (Q) for some δ > 0. A standard argument then yields b η = ∇K(ϑ) = K(ϑ)∇ϑ

a.e. in Ω × (0, T ).

Combining (5.31) and (5.32) leads to Z TZ Z K(ϑτk )∇ϑτk ) · ∇ϕ dx dt → 0

Ω

T

(5.32)

Z K(ϑ)∇ϑ · ∇ϕ dx dt

0

Ω

for every test function ϕ ∈ C0 ([0, T ]; W 2,d+ (Ω)). To complete the passage to the limit on the right-hand side of (4.50), it remains to show that ∂t χτk → χt

in L2 (0, T ; L2 (Ω)).

(5.33)

This follows from testing the discrete equation for χ (4.52) by ∂t χτk , integrating in time, and passing to the limit as k → ∞. Indeed, exploiting convergences (5.2) and (5.6)–(5.8) we deduce that Z TZ Z TZ 2 χ |χt |2 dx dt, |∂t τk | dx dt ≤ lim sup k→∞

0

Ω

0

Ω

whence (5.33). This concludes the proof of (2.54). The total energy equality (2.55) then ensues from testing (2.54) by ϕ = 1, the momentum balance (2.40) by ut , and the (pointwise) χ-equation (2.51) by χt , adding the resulting relations, and integrating in time. 6. From the p-Laplacian to the Laplacian In this Section we prove a global-in-time existence result for a suitable entropic formulation of the initialboundary value problem for system (1.1)–(1.3), in the case the p-Laplacian operator − div(|∇χ|p−1 ∇χ) is replaced by the Laplacian −∆χ, i.e. for p = 2, keeping the evolution unidirectional (i.e., µ = 1). Hence, (1.3) rewrites as χt + ∂I(−∞,0] (χt ) − ∆χ + W 0 (χ) 3 −b0 (χ) ε(u)Eε(u) + ϑ in Ω × (0, T ). (6.1) 2 We restrict, apparently for technical reasons (which however we cannot bypass), to the irreversible case µ = 1. The main idea of the technique consists in passing to the limit as δ & 0 in the following approximation of (6.1) χt + ∂I(−∞,0] (χt ) − ∆χ − δ div(|∇χ|p−1 ∇χ) + W 0 (χ) 3 −b0 (χ) ε(u)Eε(u) + ϑ 2

in Ω × (0, T ).

(6.2)

Indeed, we can apply Thm. 2.5 to the initial-boundary value problem for system (1.1)–(1.2), (6.2), with p > d (supplemented with the boundary conditions (1.4)), and conclude the existence of global-in-time entropic solutions. In this entropic formulation we will pass to the limit as δ & 0, recovering an existence result for the case p = 2. Let us now state the notion of entropic solution for the limit system as δ → 0. We mention in advance that the solution concept introduced below is weaker than the one we have obtained in the case p > d (cf. Definition (2.4)). In fact, the total energy inequality holds true only on (0, t) (cf. (6.7) below), and not on a generic interval (s, t), and so does the energy inequality in the weak formulation of the equation for χ. Moreover, the momentum equation is no longer formulated pointwise a.e. in Ω × (0, T ), but in H −1 (Ω; Rd ), a.e. in time, only. Let us also anticipate that we will confine to initial data χ0 ∈ H 1 (Ω) such that χ0 ≥ 0 a.e. in b χ0 ) ∈ L1 (Ω) as in (2.26)) and, at the same time, χ0 ≤ 1 a.e. in Ω. This and the irreversible Ω (which gives β( character of the evolution will ensure that χ ∈ [0, 1] a.e. in Ω × (0, T ), in accord with its physical meaning. Definition 6.1 (Entropic solutions to the irreversible system with p = 2). Given initial data (ϑ0 , u0 , v0 fulfilling (2.24)–(2.25), and χ0 such that χ0 ∈ H 1 (Ω),

0 ≤ χ0 ≤ 1 a.e. in Ω,

(6.3)

42

ELISABETTA ROCCA AND RICCARDA ROSSI

we call a triple (ϑ, u, χ) an entropic solution to the (initial-boundary value problem) for system (1.1)–(1.2), (6.1) with the boundary conditions (1.4), if ϑ ∈ L2 (0, T ; H 1 (Ω)) ∩ L∞ (0, T ; L1 (Ω)) ,

(6.4)

u ∈ H 1 (0, T ; H01 (Ω; Rd )) ∩ W 1,∞ (0, T ; L2 (Ω; Rd )) ∩ H 2 (0, T ; H −1 (Ω; Rd )) ,

(6.5)

χ ∈ L∞ (0, T ; H 1 (Ω)) ∩ H 1 (0, T ; L2 (Ω)),

(6.6)

(ϑ, u, χ) complies with the initial conditions (2.35)–(2.36), and with the entropic formulation of (1.1)–(1.2), (6.1) consisting of - the entropy inequality (2.37); - the total energy inequality for almost all t ∈ (0, T ]: Z tZ Z tZ E (ϑ(t), u(t), ut (t), χ(t)) ≤ E (ϑ0 , u0 , v0 , χ0 ) + g dx dr + 0

Ω

0

Z tZ h dS dr +

∂Ω

0

f · ut dx dr ,

(6.7)

W (χ) dx ;

(6.8)

Ω

where E (ϑ, u, ut , χ) :=

Z ϑ dx + Ω

1 2

Z

1 1 |ut |2 dx + e(b(χ(t))u(t), u(t)) + 2 2 Ω

Z

|∇χ|2 dx +

Ω

Z Ω

- the momentum equation utt + V (a(χ)ut ) + E (b(χ)u) + Cρ (ϑ) = f

in H −1 (Ω; Rd );

(6.9)

- the weak formulation of (6.1), viz. χt (x, t) ≤ 0 for a.a. (x, t) ∈ Ω × (0, T ), Z χt (t)ψ + ∇χ(t) · ∇ψ + ξ(t)ψ + γ(χ(t))ψ + b0 (χ(t)) ε(u(t))Eε(u(t)) ψ − ϑ(t)ψ dx ≥ 0 2 Ω for all ψ ∈

W−1,2 (Ω)

∞

∩ L (Ω),

(6.10) (6.11)

for a.a. t ∈ (0, T ),

where ξ ∈ ∂I[0,+∞) (χ) in the sense that ξ ∈ L1 (0, T ; L1 (Ω))

and

hξ(t), ψ − χ(t)iW 1,2 (Ω) ≤ 0 ∀ ψ ∈ W+1,2 (Ω) ∩ L∞ (Ω), for a.a. t ∈ (0, T ), (6.12)

as well as the energy inequality for all t ∈ (0, T ]: Z tZ Z 1 χ 2 2 χ χ |∇ (t)| + W ( (t)) dx | t | dx dr + 2 0 Ω Ω Z Z tZ 1 χ 2 χt −b0 (χ) ε(u)Eε(u) + ϑ dx dr. ≤ |∇ 0 | + W (χ0 ) dx + 2 2 0 Ω Ω

(6.13)

We are in the position now to state the main existence result of this section. Theorem 6.2 (Existence of entropic solutions, µ = 1 and p = 2). Let Ω be a bounded connected domain with Lipschitz boundary. Assume Hypotheses (I)–(III) with b0 (x) ≥ 0

for all x ∈ R,

(6.14)

and, in addition, Hypothesis (IV) (i.e., βb = I[0,+∞) ), as well as conditions (2.21)–(2.25) on the data f , g, h, ϑ0 , u0 , v0 , and (6.3) on χ0 . Then, there exists an entropic solution (in the sense of Definition 6.1) (ϑ, u, χ) to the initial-boundary value problem for system (1.1)–(1.2), (6.1), such that ξ in (6.12) is given by (2.47) and ϑ satisfies (2.48). Remark 6.3. Let us note that in Thm. 6.2 we are able to deal with the case of a Lipschitz domain Ω and we do not need C2 -regularity of Ω (2.14). The latter condition was exploited in the previous sections in order to perform the elliptic regularity estimate on u (cf. the Fifth estimate (3.21)), which is not carried out here. Indeed the regularity requirement (6.5) on u we ask for in Definition 6.1 is weaker than the one prescribed in Section 2

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43

(cf., e.g., (2.33)). Moreover, for the same reason, in this case we could also consider more general boundary conditions on u than the homogeneous Dirichlet (1.4): for example mixed Dirichlet-Neumann conditions could be taken into account. Proof. Let (ϑδ , uδ , χδ ) be a suitable family of entropic solutions to the initial-boundary value problem for (1.1)–(1.2), supplemented with initial data (ϑ0 , u0 , v0 ) fulfilling (2.24)–(2.25), and with a sequence of data (χδ0 )δ such that (χδ0 )δ ⊂ W 1,p (Ω),

0 ≤ χδ0 (x) ≤ 1 for all x ∈ Ω for all δ > 0,

χδ0 → χ0 in H 1 (Ω).

(6.15)

Observe that we cannot rigorously perform on the entropic formulation of (1.1)–(1.2) the a priori estimates in Section 3. Therefore we need to confine the discussion only to the entropic solutions which arise from the time-discretization scheme set up in Sec. 4. In the present framework (i.e. with p = 2 and µ = 1, and no upper bound on κ, cf. Hypothesis (V)), the a priori estimates for the time-discrete solutions in Prop. 4.10 are inherited in the time-continuous limit by the entropic solutions, with the exception of those corresponding to the Fifth, the Seventh, and the Eighth a priori estimates in Sec. 3, cf. also Remark 3.1. The convergences from Lemma 5.1 combined with lower semicontinuity arguments indeed ensure that the strict positivity of ϑδ (cf. (3.2)), as well as estimates (3.5), (3.14), (3.16), (3.18), (3.27), hold with constants uniform w.r.t. δ. Moreover, combining the fact that βb = I[0,+∞) with the unidirectional character of the evolution and with the fact that χδ (0) = χδ0 ∈ [0, 1] on Ω, we infer that ∃C > 0 ∀δ > 0 :

kχδ kL∞ (Q) ≤ C.

(6.16)

Therefore, repeating the compactness arguments in the proof of Lemma 5.1, based on the compactness results in [32], for every vanishing sequence δk ↓ 0 as k → ∞ there exist a not relabeled subsequence and a triple (ϑ, u, χ) along which there holds as k → ∞: ϑδk *∗ ϑ ∗

uδk * u

in L2 (0, T ; H 1 (Ω)) ∩ L∞ (0, T ; L1 (Ω)) , 2

in H (0, T ; H

−1

2

d

(Ω; R )) ∩ W 2

1,∞

(6.17) 2

d

1

1

d

(0, T ; L (Ω; R )) ∩ H (0, T ; H (Ω; R )) ,

d

(6.18)

∂t uδk → ∂t u in L (0, T ; L (Ω; R )) ,

(6.19)

χδk *∗ χ in H 1 (0, T ; L2 (Ω)) ∩ L∞ (0, T ; H 1 (Ω)) ,

(6.20)

χδk → χ

h

in L (Ω × (0, T ))

log(ϑδk ) → log(ϑ) ϑδk → ϑ

2

for all h ∈ [1, +∞) , s

(6.21)

for all s ∈ (1, 6) for d = 3 and for all s ∈ (1, +∞) for d = 2 , (6.22)

in L (0, T ; L (Ω))

in Lh (Ω × (0, T )), for every h ∈ [1, 8/3) for d = 3 and h ∈ [1, 3) if d = 2.

(6.23)

Now, in order to pass to the limit as δ & 0 we need to prove the following further convergence. Observe that, in the case of the p-Laplacian regularization for χ, we were able to prove an additional the strong convergence for ∂t u in L2 (0, T ; H 1 (Ω; Rd )). Our argument resulted from compactness arguments, relying on the Fifth a priori estimate (i.e. the elliptic regularity estimate on u). The latter is no longer at our disposal, now. Strong convergence of ∂t uδ in L2 (0, T ; H 1 (Ω; Rd )). This argument is strongly based on the irreversible character of our system. Let us test the weak formulation (2.40) of momentum equation fulfilled by the approximate solutions (ϑδk , uδk , χδk )k , by ∂t (uδk − u), where u is the limit of (uδk )k as in (6.18)–(6.19). We get Z tZ Z t 2 0= ∂tt uδk ∂t (uδk − u) dx ds + v(a(χδk )∂t uδk , ∂t (uδk − u)) ds 0

Z + 0

Ω

t

e(b(χδk )uδk , ∂t (uδk − u)) ds − ρ

0

Z tZ

Z tZ ϑδk div(∂t (uδk − u)) dx ds −

0

Ω

f ∂t (uδk − u) dx ds =: 0

Ω

5 X i=1

Ii .

44

ELISABETTA ROCCA AND RICCARDA ROSSI

Let us now deal separately with the single integrals: Z tZ Z tZ Z tZ 2 2 2 ∂tt u∂t (uδk − u) dx ds ∂tt (uδk − u)∂t (uδk − u) dx ds + ∂tt uδk ∂t (uδk − u) dx ds = I1 : = 0

0

Ω

0

Ω

1 1 = k∂t (uδk − u)(t)k2L2 (Ω;Rd ) − k∂t (uδk − u)(0)k2L2 (Ω;Rd ) + 2 2

Z

Ω

t 2 h∂tt u, ∂t (uδk − u)iH 1 (Ω;Rd ) ds ,

0

and the third integral tends to 0 when δk & 0 due to (6.18). Moreover, Z t I2 : = v(a(χδk )∂t uδk , ∂t (uδk − u)) ds 0

Z

t

=

v(a(χδk )∂t (uδk − u), ∂t (uδk − u)) ds +

0

Z

t

v(a(χδk )∂t u, ∂t (uδk − u)) ds .

0

Now, observe that in L2 (0, T ; H 1 (Ω; Rd )).

a(χδk )∂t u → a(χ)∂t u

(6.24)

This follows from the fact that a(χδk )ut → a(χ)ut and a(χδk )ε(ut ) → a(χ)ε(ut ) a.e. in Ω × (0, T ), in view of convergence (6.21) and of the continuity of a. Moreover, also due to (6.16), we have that ka(χδk )ut kH 1 (Ω;Rd ) ≤ Ckut kH 1 (Ω;Rd ) for a constant independent of k ∈ N. Therefore, using the Lebesgue theorem (6.24) ensues. This Rt 2 implies that 0 h∂tt u, ∂t (uδk − u)iH 1 (Ω;Rd ) ds tends to 0 when δk & 0, due to (6.18). Integrating by parts in time, we get Z t I3 : = e(b(χδk )uδk , ∂t (uδk − u)) ds 0 t

Z

e(b(χδk )(uδk − u), ∂t (uδk − u)) ds +

= 0

Z

t

e(b(χδk )u, ∂t (uδk − u)) ds

0

Z tZ

1 ε(uδk − u)Eε(uδk − u) dx ds + e(b(χδk (t)(uδk − u)(t), (uδk − u)(t)) b0 (χδk )∂t χδk 2 2 0 Ω Z t 1 χ − e(b( δk (0))(uδk − u)(0), (uδk − u)(0)) + e(b(χδk )u, ∂t (uδk − u)) ds , 2 0

=−

where the last integral tends to 0 (this can be shown arguing in the same way as for I2 ), while the first integral is non-negative due to the fact that ∂t χδk ≤ 0 a.e. on Ω × (0, T ) and that b0 ≥ 0.This is the point where we exploit the unidirectional character of the system (i.e. µ = 1). Finally, Z tZ Z tZ I4 := − ϑδk ε(∂t (uδk − u)) dx ds → 0 , I5 := − f ∂t (uδk − u) dx ds → 0 , 0

Ω

0

Ω

as δk & 0, due to the convergences (6.18), (6.23), as well as assumption (2.21) on f . Ultimately, we get Z t 2 k∂t (uδk − u)(t)kL2 (Ω;Rd ) + v(a(χδk )∂t (uδk − u), ∂t (uδk − u)) ds + e(b(χδk (t)(uδk − u)(t), (uδk − u)(t)) → 0 0

as δk & 0, which entails uδk → u

strongly in W 1,∞ (0, T ; L2 (Ω; Rd )) ∩ H 1 (0, T ; H 1 (Ω; Rd )) .

(6.25)

Conclusion of the proof. Using this strong convergence, we can now pass to the limit as k → ∞ in the energy inequality (2.42) in the weak formulation of the equation for χδk as follows. We have to identify the weak limit of + ε(uδk (x, t))E(x)ε(uδk (x, t)) ξδk (x, t) = −Iχδ =0 (x, t) γ(χδk (x, t)) + b0 (χδk (x, t)) − ϑδk (x, t) . (6.26) k 2 First of all note that (Iχδ =0 )k is bounded in L∞ (Q) independently of k ∈ N. Hence, we can select a subsequence k (Iχδ =0 )k weakly star converging in L∞ (Q) to some I. Observe that we cannot establish that I = Iχ=0 . On k

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45

ε(u )Eε(u ) the other hand, it follows from the previously proved convergences that (γ(χδk ) + b0 (χδk ) δk 2 δk − ϑδk )+ strongly converges in L1 (Q) to (γ(χ) + b0 (χ) ε(u)Eε(u) − ϑ)+ . Hence we identify 2

ε(u(x, t))Eε(u(x, t)) − ϑ(x, t))+ (6.27) 2 and observe that ξδk * ξ in L1 (Q). Then, integrating (2.42)δk from 0 to T and passing to the limit as k → ∞, using the fact that or all ψ ∈ Lp (0, T ; W−1,p (Ω)) ∩ L∞ (Q) Z Z T p−1 δk |∇χδk |p−2 ∇χδk · ∇ψ dx dt ≤ δk k∇χδk kL p−1 (Q;Rd ) k∇ψkLp (Q;Rd ) → 0 , 0 Ω ξ = −I(x, t)(γ(χ(x, t)) + b0 (χ(x, t))

we get Z TZ

Z TZ χt (t)ψ + ∇χ(t) · ∇ψ + γ(χ(t))ψ + b0 (χ(t)) ε(u(t))Eε(u(t)) ψ − ϑ(t)ψ dx dt ≥ − ξ(t)ψ dxdt , 2 0 Ω 0 Ω (6.28) for all ψ ∈ Lp (0, T ; W−1,p (Ω)) ∩ L∞ (Q), where ξ is defined in (6.27). From (6.28), we get (6.11). It remains to show that χ complies with the variational inequality (6.12). To do so, we have to pass to the limit in (2.43)δk , whence Z T Z ξδk (ψ − χδk (t)) dx ζ(t) dt ≥ 0 for all ψ ∈ W+1,p (Ω)L∞ (Ω) and all ζ ∈ L∞ (0, T ) with ψ, ζ ≥ 0. 0

Ω

Observe that the two weak convergences χδk *∗ χ in L∞ (Q) and ξδk * ξ in L1 (Q) do not allow for a direct RR limit passage in the term Q ξδk χδk ζ dx dt, which equals zero for all k ∈ N due to (6.26). Indeed, we need to argue in a more refined way. It follows from (6.21) that χδk converges almost uniformly to χ in Q, i.e. for every > 0 there exists Q ⊂ Q such that |Q \ Q | < and χδk → χ uniformly on Q . The latter property implies that I ≡ 0 on Q ∩ {Iχ=0 ≡ 0} . (6.29) ¯ χ χ χ Indeed, Iχ=0 (x, t) = 0 implies (x, t) 6= 0. Since δk converges to uniformly on Q , there exists an index k, ¯ χ independent of (x, t), such that for all k ≥ k, δk (x, t) 6= 0, hence Iχ =0 (x, t) = 0. With this argument we δk

conclude that that Iχδ =0 ≡ 0 on Q ∩ {Iχ=0 ≡ 0}, whence (6.29). It follows from (6.29) and (6.27) that k ZZ χ ξ(x, t) (x, t) = 0 for a.a. (x, t) ∈ Q , whence ξ(x, t)χ(x, t)ζ(t) dx dt = 0 . Q

On the other hand, using the properties of the Lebesgue integral we have that ZZ ∀ η > 0 ∃ = η > 0 : |Q \ Q | < ⇒ |ξ(x, t)χ(x, t)ζ(t)| dx dt < η. Q\Q

Therefore we conclude that

Z Z ξ(x, t)χ(x, t)ζ(t) dx dt < η,

∀η > 0

Q

i.e.

ZZ

ξ(x, t)χ(x, t)ζ(t) dx dt = 0 = lim

ZZ

k→∞

Q

ξδk χδk ζ dx dt Q

Hence ZZ

ξδk (ψ − χδk )ζ dx dt →

0≤ Q

ZZ

ξ(ψ − χ)ζ dx dt =

Q

Z 0

T

Z

χ ξ(ψ − (t)) dx ζ(t) dt,

Ω

which implies Z

ξ(t)(ψ − χ(t)) dx ≥ 0

for a.e. t ∈ (0, T )

Ω

With a density argument we get (6.12) for all ψ ∈ W+1,2 (Ω).

for all ψ ∈ W+1,p (Ω) ∩ L∞ (Ω).

46

ELISABETTA ROCCA AND RICCARDA ROSSI

Convergences (6.17)–(6.23) also guarantee the passage to the limit in the momentum equation, whence (6.9). Finally, we pass to the limit in the entropy inequality (2.37) and in the total energy inequality (2.38) by the very same compactness/lower semicontinuity arguments as in the proof of Theorem 2.5, thus deducing (2.37) and the total energy inequality (6.7) on the generic interval (0, t). Remark 6.4. Notice that, we have been able to obtain the energy inequalities (6.13) and (6.8) only on intervals of the type (0, t), and not on the generic interval (s, t) ⊂ (0, T ), due to the weak convergence of (∇χδk ) in L2 (Q; Rd ), which does not yield the pointwise-in-time convergence required to take the limit of the right-hand sides of (2.44) and (2.38). It is an open poblem to improve the convergence of (∇χδk ) to a strong one. This limit passage also reveals that the notion of entropic solution enjoys stability properties. It is clearly the right one in the present framework, and, seemingly, the entropy inequality cannot be improved to an equality, at least with these techniques. Indeed, due to a lack of elliptic regularity estimates on the displacement which were previously made possible by the p-Laplacian regularization, in the limit as δ ↓ 0 the right-hand side of the heat equation is only estimated in L1 (Q). References [1] G. Akagi, Global attractors for doubly nonlinear evolution equations with non-monotone perturbations, J. Differential Equations 250 (2011), 1850–1875. [2] M. Bul´ıcek, E. Feireisl, J. M´ alek: A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Analysis: Real World Applications 10 (2009), 992–1015. [3] H. Brezis: Op´ erateurs Maximaux Monotones et S´ emi-groupes de Contractions dans les Espaces de Hilbert, North-Holland Mathematics Studies, no. 5., North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [4] P. Ciarlet: Mathematical elasticity. Vol. I. Three-dimensional elasticity, Studies in Mathematics and its Applications, 20, North-Holland Publishing Co., Amsterdam, 1988. [5] P. Colli, A. Visintin: On a class of doubly nonlinear evolution equations, Comm. Partial Differential Equations 15 (1990), 737–756 [6] M. Eleuteri, E. Rocca, G. Schimperna: On a non-isothermal diffuse interface model for two-phase flows of incompressible fluids, preprint arXiv:1401.3244 (2014), 1–26. [7] E. Feireisl: Mathematical theory of compressible, viscous, and heat conducting fluids, Comput. Math. Appl. 53 (2007), 461–490. [8] E. Feireisl, M. Fr´ emond, E. Rocca, G. Schimperna: A new approach to non-isothermal models for nematic liquid crystals, Arch. Rational Mech. Anal. 205 (2012), 651–672. [9] E. Feireisl, H. Petzeltov´ a, E. Rocca: Existence of solutions to a phase transition model with microscopic movements, Math. Methods Appl. Sci. 32 (2009), 1345–1369. [10] E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential, Comm. Math. Sci., 12 (2014), 317–343. [11] E. Feireisl, E. Rocca, G. Schimperna, A. Zarnescu, Nonisothermal nematic liquid crystal flows with the Ball-Majumdar free energy, preprint arXiv:1310.8474 (2013), 1–27, WIAS Preprint No. 1865, (2013). [12] M. Fr´ emond: Non-smooth thermomechanics, Springer-Verlag, Berlin, 2002. [13] M. Fr´ emond: Phase Change in Mechanics, Lecture Notes of the Unione Matematica Italiana 13 Springer-Verlag, Berlin, 2012. ´ [14] P. Germain: Cours de m´ echanique des milieux continus, Tome I: Th´ eorie g´ en´ erale. Masson et Cie, Editeurs, Paris, 1973. [15] M. Grasselli, A. Miranville, R. Rossi, G. Schimperna, Analysis of the Cahn-Hilliard equation with a chemical potential dependent mobility, Comm. Partial Differential Equations 36 (2011), 1193–1238. [16] C. Heinemann, C. Kraus: Existence results of weak solutions for Cahn-Hilliard systems coupled with elasticity and damage: Adv. Math. Sci. Appl. 21 (2011), 321–359. [17] C. Heinemann, C. Kraus: Existence results for diffuse interface models describing phase separation and damage: European J. Appl. Math. 24 (2013), 179–211. [18] A.D. Ioffe: On lower semicontinuity of integral functionals, SIAM J. Control Optimization 15 (1977), 521–538. [19] P. Krejˇ c´ı, E. Rocca: Well-posedness of an extended model for water-ice phase transitions, Discrete Contin. Dyn. Syst. Ser. S 6 (2013), 439–460. [20] J.-L. Lions, E. Magenes: Non-homogeneous boundary value problems and applications, Vol. I., Springer-Verlag, New YorkHeidelberg, 1972.

A PDE SYSTEM FOR PHASE TRANSITIONS AND DAMAGE IN THERMOVISCOELASTICITY

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[21] M. Marcus, V. Mizel: Every superposition operator mapping one Sobolev space into another is continuous, J. Funct. Anal. 33 (1979), 217–229. [22] A. Mielke, F. Theil: On rate-independent hysteris models, Nonlinear Diff. Eq. Appl. 11 (2004), 151–189. ´ [23] Neˇ cas J.: Les m´ ethodes directes en th´ eorie des ´ equations elliptiques. Masson et Cie, Editeurs, Paris, 1967. [24] L. Nirenberg: On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa (3) 13 (1959), 115–162. [25] R.H. Nochetto, G. Savar´ e, C. Verdi: A posteriori error estimates for variable timestep discretizations of nonlinear evolution equations, Comm. Pure Appl. Math. 53 (2000), 525–589. [26] E. Rocca, R. Rossi: Analysis of a nonlinear degenerating PDE system for phase transitions in thermoviscoelastic materials, J. Differential Equations 245 (2008), 3327–3375. [27] E. Rocca, R. Rossi: Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math. 53 (2008), 485–520. [28] E. Rocca, R. Rossi, A degenerating PDE system for phase transitions and damage, preprint arXiv:1205.3578v1 (2012), 1–53. In press on Math. Models Methods Appl. Sci. [29] T. Roub´ıˇ cek: Nonlinear Partial Differential Equations with Applications, Birkh¨ auser, Basel, 2005. [30] T. Roub´ıˇ cek: Thermodynamics of rate independent processes in viscous solids at small strains, SIAM J. Math. Anal. 40 (2010), 256–297. [31] G. Savar´ e: Regularity results for elliptic equations in Lipschitz domains, J. Funct. Anal. 152 (1998), 176–201. [32] J. Simon: Compact sets in the space Lp (0, T ; B), Ann. Mat. Pura Appl. 146 (1987), 65–96. [33] J. Zimmer: Global existence for a nonlinear system in thermoviscoelasticity with nonconvex energy, J. Math. Anal. Appl. 292 (2004), 589–604. Elisabetta Rocca, Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, D-10117 Berlin, ` di Milano, Via Saldini 50, I-20133 Milano, Italy Germany, and, Dipartimento di Matematica, Universita E-mail address: [email protected] and [email protected] ` di Brescia, Via Valotti 9, I-25133 Brescia, Italy Riccarda Rossi, DICATAM - Sezione di Matematica, Universita E-mail address: [email protected]

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